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PacificJournal ofMathematics
IN THIS ISSUE—
Tom M. (Mike) Apostol, On the Lerch zeta function . . . . . . . . . . . . . . . 161Ross A. Beaumont and Herbert S. Zuckerman, A characterization of
the subgroups of the additive rationals . . . . . . . . . . . . . . . . . . . . . . . 169Richard Bellman and Theodore Edward Harris, Recurrence times for
the Ehrenfest model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179Stephen P.L. Diliberto and Ernst Gabor Straus, On the
approximation of a function of several variables by the sum offunctions of fewer variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Isidore Isaac Hirschman, Jr. and D. V. Widder, Convolutiontransforms with complex kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Irving Kaplansky, A theorem on rings of operators . . . . . . . . . . . . . . . . 227W. Karush, An iterative method for finding characteristic vectors of
a symmetric matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233Henry B. Mann, On the number of integers in the sum of two sets of
positive integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249William H. Mills, A theorem on the representation theory of Jordan
algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255Tibor Radó, An approach to singular homology theory . . . . . . . . . . . . . 265Otto Szász, On some trigonometric transforms . . . . . . . . . . . . . . . . . . . . 291James G. Wendel, On isometric isomorphism of group algebras . . . . 305George Milton Wing, On the L p theory of Hankel transforms . . . . . . . 313
Vol. 1, No. 2 December, 1951
PACIFIC JOURNAL OF MATHEMATICS
EDITORS
HERBERT BUSEMANN R. M. ROBINSONUniversity of Southern California University of CaliforniaLos Angeles 7, California Berkeley 4, California
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UNIVERSITY OF CALIFORNIA PRESS BERKELEY AND LOS ANGELES
COPYRIGHT 1951 BY PACIFIC JOURNAL OF MATHEMATICS
ON THE LERCH ZETA FUNCTION
T. M. APOSTOL
l Introduction. The function φ(x9a9 s), defined for Hs > 1, x real, a ψ nega-
tive integer or zero, by the series
o° 2nπix
(1.1) φ (x,ass) = Σ 1 7 y '
was investigated by Lipschitz [4; 5], and Lerch [3]. By use of the classic
method of Riemann, φ{x, α, s) can be extended to the whole s-plane by means of
the contour integral
1 Λ zs"1eaz
α.2) «•".•)/where the path C is a loop which begins at —-00 , encircles the origin once in the
positive direction, and returns to — 00 . Since I(x9a, s) is an entire function of s,
and we have
d 3 ) φ(x,a,s)=Γ{l-s)l(x,a,s),
this equation provides the analytic continuation of φ. For integer values of x,
φ(x,a,s) is a meromorphic function (the Hurwitz zeta function) with only a simple
pole at s — 1. For nonintegral x it becomes an entire function of s. For 0 < x < 1,
0 < a < 1, we have the functional equation
(1-4) φ{x, a,l-s)
first given by Lerch, whose proof follows the lines of the first Riemann proof of
the functional equation for ζ(s) and uses Cauchy's theorem in connection with the
contour integral (1.2).
Received March 4, 1951.
Pacific J. Math. 1 (1951), 161-167.
161
162 T. M. APOSTOL
In the present paper, §2 contains a proof of (1.4) based on the transformation
theory of theta-functions. This proof is of particular interest because the usual
approach (Riemann's second method) does not lead to the functional equation
(1.4) as might be expected but to a different functional relationship (equation (2.4)
below). Further properties of φ(x9α,s), having no analogue in the case of ζ(s),
are needed to carry this method throu'gh to obtain (1.4).
In §3 we evaluate the function φ(xf α, s) for negative integer values of s .
These results are expressible in closed form by means of a sequence of functions
βn{a9e2πix) which are polynomials in a and rational functions in e2Ίίιx. These
functions are closely related to Bernoulli polynomials; their basic properties also
are developed here.
2. Functional Equation for φ(x9 a, s ) . The theta-f unction
00
^3 (y lτ) = Σ exp (πin2r + 2iny)n=-oo
has the transformation formula [6,p.475]
If we let
θ(x,a, z) = exp(—ττa2z)&3(πx+πiaz\iz) = Σ exp(2nπix — πz(a + n)2) ,n=-oo
then we have the functional equation
(2.1) θ(a,~x,l/z)= [exp(2ττiax)]zi/2θ(χ,a,z).
The key to Riemann's second method is the formal identity
(2.2) π-°/*Γ(s/2) Σ «n/n's/2 = Γ ^ " 1 Σ α nexp(-τ,z/ n)dz.n = l
Taking first an = exp[(2mU ~l)x)] , fn = U - 1 + a)2 in (2.2), and then
an = exp(— 2τrinx), fn — (n — a)2 , we obtain
ON THE LERCH ZETA FUNCTION 163
(2.3) 7 τ - s / 2 Γ ( s / 2 ) {φ{x,a,s) + exp(-2πίx) φ(-x,l - a,s)}
In the second integral in (2.3) we apply (2.1) and replace z by 1/z. Denoting the
expression in braces by K(x9α9s), replacing 5 by 1 — s, re by ~ o , α by x, using
θ{ — α,x9 z) — θ(αf ~xf z), and the relation
( ^ p ) 2(2τrΓs
c o s (τrs/2)Γ(s) ,
we are led to
(2.4) Λ(x,α,l - s) = 2(2τ7)"5 cos (ττs/2)Γ(s) exp(~2πίαx) A ( - α , x,s) .
Thus Riemann's method gives us a functional equation for Λ instead of (1.4). At
this point we introduce the differential-difference equations satisfied by φ ,
namely:
α,s) , ,= —sφ\x,α,s + 1;
σα
and
(2.6) h 2τfiaφ\xiays) — 2τri φ{x, α, s — l) .
'σx
The first of these follows at once from (1.1). To obtain (2.6) we first write
exp[2πi(n + a)x~\φ(x,α,s) = exp( — 2πiαx)
(n + α)s
before differentiating with respect to x. The equations hold for all s by analytic
continuation.
The proof of (1.4) as a consequence of (2.4) now proceeds as follows. We differ-
entiate both sides of (2.4) with respect to the variable α, using (2.5) on the left
and (2.6) on the right, and replace s by s + 1 in the resulting equation. This
164 T. M. APOSTOL
leads to the relation
Φ{x, α, 1 —s) —exp(— 2ττix) φ(— x,\ —α, 1 —s)
= 2i(2π)"s sin (πs/2)Γ(s)
X [exp(~2τriαΛ:)φ(— α, %, S ) — exp(~ 2πia(l— x))φ(a, 1— x,s)] .
Adding this equation to (2.4) gives the desired relation (1.4).
This method has already been used by N. J. Fine [ l ] to derive the functional
equation of the Hurwitz zeta function. Fine's proof uses (2.5) with x = 0. In our
proof of (1.4) it is essential that x φ 0 since we have occasion to interchange the
variables x and α, and φ{x9a,s) is not regular for a — 0 hence Fine's proof is
not a special case of ours. Furthermore, putting x = 0 in (1.4) does not yield the
Hurwitz functional equation, although this can be obtained from (1.4) as shown
elsewhere by the author.
3. Evaluation of φ{x, a, — n). If x is an integer, then φ{x, a9 s) reduces to the
Hurwitz zeta function ζ(s,a) whose properties are well known [6,pp.265-279] .
For nonintegral x the analytic character of φ is quite different from that of ζ ( s , α ) ,
and in what follows we assume that x is not an integer.
The relation (2.6) can be used to compute recursively the values of φ(x, α, s)
for s = —1, —2, — 3, . As a starting point we compute the value at s = 0 by
substituting in (1.2). The value of the integral reduces to the residue of the
integrand at z = 0 and gives us
Φ(x,a,0) = -. - r = (i/2) cot πx + 1/2 .1 "" exp(27T ix )
Using (2.6) we obtain
0 ( * , α , - l ) = (α/2)(i cot 77* + l) - (l/4) csc2πx ,
φ(x,a,-2) = (α2/2)(i cot πx + 1/4) - {a/2) csc2rrx - (i/4) cot πx csc2πx .
If we put 5 = — 71 in (1.2) and use Cauchy's residue theorem we obtain, for
n > 0, the relation
φ\x9a,—n) =
ON THE LERCH ZETA FUNCTION 165
where βn{a, α) is defined by the generating function
(3.1) , _ί^_. j«£«i . . .α e ' - l „.„ »!
When OC = 1, βn(af(χ) is the Bernoulli polynomial Bn(a) For our purposes we
assume Ot ψ 1, and in the remainder of this section we give the main properties of
the functions βn(a,(X).
Writing βn(a) instead of βn(0, α) we obtain from (3.1):
(3.2) βn(a, α) = £ lϊ) βkfa) an~k (n > 0) ,
from which we see that the functions βn(a9 OC) are polynomials in the variable α.
The defining equation (3.1) also leads to the difference equation
(3-3) aβn(a + l,α) -βn(a,a) =nan~1 (n > l) .
Taking α = Owe obtain, for n — 1, the relation
(3.4) 0LβΛl,0i) = 1 +j81(α)
while for ^ > 2 we have
(3.5) α/3 B ( i ,α)=y8 B (α) .
Putting α — 1 in (3.2) now allows us to compute the functions βn(CL) recursively
by means of
(3.6) /3n(l,α) = Σ (n\βk{o>)fe=o ' '
and (3.4), (3.5). From (3.1) we obtain /30(α) = 0; the next few functions are found
to be:
3α(α
u - 1 (cc- l ) 2 (&.-1)
_ , , 4α(ct2 + 4α + l) 5α(α3 + llα2 + llα + l)β4(a) = — , /35(α ) = —
( α - 1 ) 4 (α — l ) 5
166 T. M. APQSTOL
6α(α4 + 26α3 + 66α2 + 26α
(α-l)6
The general formula is
n-1(3.7)
where &jj are Stirling numbers of the second kind defined by
J 0 f e
i I
with
0 n = ( Δ ^ Λ ) X = O , Δ > 0 π = 0 i f j > n , Δ ° 0 ° = 1 ,
in the usual notation of finite differences. (A short table of Stirling numbers is
given in [2].)
To prove (3.7) we put
QLed α - 1α
1 - α/
Using Herschel ' s theorem [2, p . 73] which expresses (ez — 1) Λ a s a power
ser ies in z we obtain
00 m
α I s Q I
Comparing with
.α)= Σ βn(a)~nl
we get (3.7).
The following further properties of the numbers βn{a90L), which closely re-
semble well-known formulas for Bernoulli polynomials, are easy consequences of
ON THE LERCH ZETA FUNCTION 167
the above:
( 0 < p < n ) ,
k=o
J Pn(t,CC)cit =α
lnλβk(a,a)b»-k,
Jα
n + 1
Taking a — b ~ 1 and using (3.3), we can also use this last equation to obtain
the functions βn(a, a ) recursively by successive integration of polynomials.
As a final result, taking a = 0,1,2, ,m — 1 in(3.3)and summing we obtain
U.8) Σ « = " T Γ Σ ^n+i (α. α) + — .
a generalization of the famous formula giving Σan in terms of Bernoulli poly-
nomials. This result is somewhat surprising because of the appearance of the
parameter α on the right. (When α = 1, (3.8) reduces to the Bernoulli formula.)
R E F E R E N C E S
1. N. J. Fine, Note on the Hurwitz zeta-Function, Proc. Amer. Math. S o c , scheduledto appear in vol. 2 (1951).
2. T. Fort, Finite Differences, Clarendon Press, Oxford, 1948.
oo '2kπiχ3. M. Lerch, Note sur la function ϋ(w,xts) = £ — , Acta Math. 11 (1887),
19-24. *=o (" + *>
4. R. Lipschitz, Untersuchung einer aus vier Eίementen gebildeten Reihe, J. ReineAngew. Math. 54 (1857), 313-328.
5. , Untersuchung der Eigenschaften einer Gattung von unendlichen Reihen,J. Reine Angew. Math. 105 (1889), 127-156.
6. E. T. Whittaker and G. N. Watson, Modern Analysis, University Press, Cambridge,England, 1945.
CALIFORNIA INSTITUTE OF TECHNOLOGY
A CHARACTERIZATION OF THE SUBGROUPS OF THE
ADDITIVE RATIONALS
Ross A. BEAUMONT AND H. S. ZUCKERMAN
1. Introduction. In the class of abelian groups every element of which (except
the identity) has infinite order, the subgroups of the additive group of rational
numbers have the simplest structure. These rational groups are the groups of rank
one, or generalized cyclic groups, an abelian group G being said to have rank one
if for any pair of elements, a ^ 0, b ^ 0, in G, there exist integers m, rc, such that
ma — nb ψ 0. Although many of the properties of these groups are known [ l ] , it
seems worthwhile to give a simple characterization from which their properties
can easily be derived. This characterization is given in Theorems 1 and 2 of §2,
and the properties of the rational groups are obtained as corollaries of these
theorems in §3. In §4, all rings which have a rational group as additive group are
determined.
Let pi, p 2 > , pi, * * be an enumeration of the primes in their natural
order; and associate with each pj an exponent kj, where kj is a nonnegative inte-
ger or the symbol °°. We consider sequences i; kγ , k2 > * * , λy, , where i is
any positive integer such that (i,py) = 1 if kj > 0, and define {i; kγ , k29* * •>
kj9 •) = (ι; kj) to be the set of all rational numbers of the form ai/b, where a
is any integer and b is an integer such that b — Ώp.pJJ with τij < kj> Then each
sequence determines a well-defined set of rational numbers. The symbol Π desig-
nates a product over an arbitrary subset of the primes that satisfy whatever condi-
tions are put on them; Π designates a product over all primes that satisfy the
given conditions.
2 Characterization of the rational groups. We show that the nontrivial sub-
groups of R are exactly the subsets (i kj) defined in the introduction.
THEOREM 1. The set (i;kj)is a subgroup o//?+, the additive group of rational
numbers. We have (i kj) — (i'; k 'j) if and only if i — i', kj = k y for all /.
Received October 11, 1950. Presente4 to the American Mathematical Society, JuneΓ7, 1950.
Pacific J. Math. 1 (1951), 169-177.
169
170 ROSS A. BEAUMONT AND H. S. ZUCKERMAN
Proof. If ai/b £ (i kj), ci/d £ (i kj), then b = H^pp, d = Π ^ . p J ^
[ i , rf] — Ώp.psJ, where SJ — max {ΠJ, mj) < kj» Writing [b, d] = bb' = dd', we
have
ai ci _ b1 ai d'ci __ (6'α ~ d1 c)i r~ ( ^ \
7 ~ 7 ~ ΐ ΰ Π "EMJ ~ ΪMΪ e U; jh
It is clear that different sequences determine different subgroups.
In the sequel we need the following properties of a subgroup G ψ 0 of R+.
(1) Every ζ £ G has the form ζ = ai/b, {ai, b) - 1, where i is the least
positive integer in G.
For every ζ we have ζ = m/b, where (m, b) = 1; and if i is the least positive
integer in G, then m = αi + r and m — ai C G imply r = 0.
(2) If αί/6 C G, ΐ C G, and (α, 6) - 1, then i/b C G.For there exist integers k, I such that ka + Zό = 1 and
(3) If ai/b C G where i is the least positive integer in G, and (α, b) = 1,
then (i, i) = 1.
By (2), t/fc C G; and if (£, 6) ^ 1, h/b' C G with A < ί. Then b' (h/b') -
h C G.
We assume in the proof of the remaining properties that the elements of G are
written in the canonical form ai/b with {ai, b) — 1 and i the least positive integer
in G.
(4) If αi/6c C G, then i/6 C G.
For cαΐ/όc = α£/6 C G and ί/i C G by (2).
(5) If ai/b, ci/d- € G, and if {b, d) = 1, then i/M £ G. For by (2) we haveFor by (2) we have
i (kb + ld)i ki li
6d 6d d b
TίfEOREM 2. If G ^ 0 is a subgroup of /?+, ί/ie i iAere exists a sequence
{i; kι , k2 , ' ' *, kj, •) such that G — {i; kj).
Proof. By (1), every ζ £ G has the form ζ = αι/6, {ai, b) = 1, where i is the
A CHARACTERIZATION OF THE SUBGROUPS OF THE ADDITIVE RATIONALS 171
least positive integer in G. We write all elements of G in this form. If, for every
I, there exist ai/b £ G such that p . \b, let Ay — °°. If not, let Ay — max A such
that p . I b for some ai/b £ G. Since (ai, b) = 1, we have (ί, py) = 1 if Ay > 0. By
the definition of i and kj, G is contained in (ΐ Ay). Now every element of {i; kj)
has the form ai/{pn^ p* Γ ) , where ΠJ < kj and (a, p " * p " r ) = 1. By (4) and
the definition of Ay, G contains every i/p^J with ΠJ < kj, and by(5), G contains
<u/(p?1 p? r). Hence G = (*; Ay).
3 Properties of the rational groups* In this section, properties of the rational
groups are obtained as corollaries of the theorems of § 1 .
C O R O L L A R Y 1 . The g r o u p ( i ; kj) i s a s u b g r o u p of ( i f A y ) if a n d o n l y if
j £ A yA y and i = mi'
COROLLARY 2. The group (i; kj) is cyclic if and only if kj < °° for all j and
kj = 0 for almost all j .
Proof. If (i Ay) is cyclic, it is generated by ai/b with {ai, b) = 1. Since every
element of (i Ay) has the form nai/b, we have a — 1 and 6 = Π t > 0 p .J. Con-
versely (i Ay) contains ι / Π ^ > 0 pW, and this element generates (i; kj).
COROLLARY 3. We have (£; Ay) = (*'; Ay) if and only if both kj = Ay for
almost all j , and, whenever kj ψ Ay, both are finite. Every isomorphism between
(i; kj) and (i*; Ay) is given by
αi
b
where
- = Π'
mαi
n6
πfinite
Πfinite
Proof. If (i; kj) = (i1; Ay), then i —> m ΐ ' A with (mι ;, n) = 1. If
then m7) —> mi' and ni —» mi', so that mrj = ni, or τ\ — ni/m.
172 ROSS A. BEAUMONT AND H. S. ZUCKERMAN
Hence ni/m —» i1. We write
then for nj < kj we have
Pί\~• paJ
while for n's < Ay we have
We have the following alternatives with consequences which follow from (3):
I. j = a i : nj - k'j < α/ < kj - n'j
II. j = ^« : «) "" fe; < K 1 *> ~ Λj
III. •" ^j < fey i n'j <
It follows that kj = °° implies k'j = °° and conversely. With both Ay and A y
finite we may choose ray = Ay and n y = A y and we have:
I.
II.
III.
= fe; - fey
= fey — fey
J Φ : fey = fey
We have Ay = A y if and only if / φ Oί/, / 'φ βm. In particular, we have Ay = A y
for almost all j . If Ay > A 'y, then j = OC/ and α/ = Ay — A 'y. If A y > Ay, then j — βm
and 6 m = A y — Ay.
Now i —> mi'/n implies αι/ό —> ami'/bn, so that the only isomorphisms
between (i; Ay) and (ί 1; Ay) are those described in the corollary. Incidentally, we
A CHARACTERIZATION OF THE SUBGROUPS OF THE ADDITIVE RATIONALS 1 7 3
have derived necessary conditions for the relation {i kj) = W; k'j)
With the necessary conditions satisfied, we check that the given correspond-
ence actually is an isomorphism. These conditions imply that the correspondence
is single-valued with a single-valued inverse from (i; kj) onto (i1; k'j). It is clear
that addition is preserved.
COROLLARY 4. The group (i; kj)admits only the identity automorphism if and
only if kj is finite for all j .
Proof. If kj is finite for all j , we have by Corollary 3, with kj — k'j for all / ,
that m — n = 1. Conversely, if any kj — °°, then the correspondence of Corollary 3
gives us nontrivial automorphisms.
The multiplicative group of the field of rational numbers, /?*, is a direct pro-
duct of the infinite cyclic subgroups of Rx generated by the prime numbers p, for
all k. Such a subgroup consists of the elements p, , p? , , 1, 1/W > ^/p&> * * *
COROLLARY 5. The group of automorphisms of (i; kj) is isomorphic to the
direct product of all of the infinite cyclic subgroups of Rx generated by those
primes p^ for which kj = °°.
Proof. By Corollary 3, there is a (1—1) correspondence between the automor-
phisms of (i; kj) and the rational numbers U/N with (M, N) = 1, where M and N
are arbitrary products of those primes for which kj = °°. This correspondence
clearly preserves multiplication and the set of all rationale M/N has the stated
structure as a group with respect to multiplication.
COROLLARY 6. For any two subgroups (i; kj) and (£'; k'j) of /?+, the set T
consisting of all ordinary products of an element of (i kj) with an element of
(i'; k 'j) is again a subgroup of R+.
Proof. We have T = (1 Kj), where
with Sj = minίoty, k'j) + min(θty, kj), where (Xy is the highest power of pj that
divides i, and Oί'y the highest power of py that divides i'.
COROLLARY7. // ( i ; kj) > (i ;; k'j) and ph' is the maximum power of pj such
that p.i divides i '/i , then the difference group (i kj) — (i1; k'j) is a direct sum
174 ROSS A. BEAUMONT AND H. S. ZUCKERMAN
of the groups Gj where
(i) Gj is the cyclic group,
«';*;>
Pri}
if kj is finite;
(ii) Gj is the group of type pc
(•';*}: J ΌR)
if kj is infinite and k'j is finite;
(iii) Gj — [θ] ifkj = k'j = 00,
4. Rings which have a rational group as additive group. The distributive laws
in any ring S with (i kj) as additive group are used to determine all possible
definitions of multiplication in S.
LEMMA. If S is a ring with additive group (i kj), then multiplication in S is
defined by
ai ci ac , N— X — = — 1 X 1 .6 d bd
Proof. We prove this by showing that
lai ci\ , xbd[— X— = ac (i X i) .
\b d
We have
ac (i X i) = ai X ci (by the distributive laws in S)
[ αi ail Γci ci
— + • • • + — X — + • • • + —6 6 J Id d
b summands d summands
A CHARACTERIZATION OF THE SUBGROUPS OF THE ADDITIVE RATIONALS 1 7 5
whence ac (i X i )
[ ai ci] \ai ci] ,, , ,. ., . , . o x
— X — + + — X — (by the distributive laws in S)b d\ Lj 16 d\
bd summands
ai ciΛ
6 d
THEOREM 3. // there is an infinite number of kj such that 0 < kj < °°, then
the only ring S with (i kj) as additive group is the null ring. If 0 < kj < °° for
only a finite number of kj, then S is a ring with additive group (i kj) if and only
if multiplication in S is defined by
bd Π' pp
where A ' and nj are arbitrary.
Proof. If S is a ring with additive group (i kj), then i X i = Ai/B £ (i;kj)9
where (Ai, B) = 1, B = Π p W, nj < kj. By the lemma, we have
ai ci acAί
b d ~ bdB
If 0 < kr < °°, this yields in particular
i i Ai
Therefore (pΓ, B) = 1, for otherwise we would have 2kr + nr < kΓ9 which is im-
possible. Hence, B = Π p?J is a product of primes for which kj = °°, and it is
necessary that p^ΓU. If there is an infinite number of primes py with 0 < kj < °°,
then A — 0 and (ai/b) X (ci/d) = O This proves the first statement in the theorem.
If 0 < kj < °° for only a finite number of primes p. , then
A = A' Π p)i
176 ROSS A.BEAUMONT AND H. S. ZUCKERMAN
Together with what has been proved above, this gives
acA' j Π Pjai ci _ \o<kj<co
X — • ,
b d _ , '
bd Π P?}
j
where A ' and nj > 0 are arbitrary integers.
Conversely, this definition of multiplication always makes (i; kj) a ring. Closure
with respect to X is insured by providing p .3 in the numerator when 0 < kj <°°,
and the associative and distributive laws are readily verified.
COROLLARY l The set {i; kj) is a subring of R if and only if there is no
kj such that 0 < kj < °°.
Proof. Let (i kj) be a subring of R and assume that for at least one kj we
have 0 < kj < °°. If 0 < kj < °° for infinitely many kj, then (ί kj) is not a subring
of R, since by Theorem 3 it is the null ring. If 0 < ky < °° for a finite number of
kj, then multiplication in any ring with (i kj) as additive group is given by the
formula of the theorem. Hence this must reduce to ordinary multiplication for some
choice of A ' and nj that is,
A' Π r J
Π'pW= i; A' Π p / ' = i Π ' P ; ι .
By hypothesis, at least one pj with kj > 0 appears in the left member of the above
equality. Since no prime appears in both products, we have pj\i. This contradicts
(i, pj) = 1 for kj > 0.
Conversely, let every kj be either 0 or oo, By the theorem, we have
a t
6X
Cl
dbd
acΛ i
rr γ
and we may select A1 = i, Π ^.=00 pni ~ 1, yielding ordinary multiplication.
C O R O L L A R Y 2 . // ( i ; kj) is a subring of R , then ( i ; kj) is a ring under the
A CHARACTERIZATION OF THE SUBGROUPS OF THE ADDITIVE RATIONALS 1 7 7
multiplication
ax ci ac lei
b d ~ bd \ f
for arbitrary ei/f £ {i kj).
Proof. By Corollary 1, we have kj ~ 0 or kj = °°, so that every element of
(i kj) has the form
A'i
ir $*
and by the theorem these are just the multipliers which are used to define multi-
plication.
COROLLARY 3. If S is a ring with additive group {i; kj), then either S is a
null ring or S is isomorphic to a subring of R,
Proof. If S is not null, the correspondence
ai aA ai ci acAi— —> — - , where — X — = ,6 bB b d bdB
is (1~1) from S on a subset of R, and
ai ci (da + bc)i (da + bc)A __ aA cA
b d bd bdB bB dB '
ai ci acAi acA2 __ aA cA
b d ~ bdB bdB2 ~ bB dB
COROLLARY 4. All rings with additive group Λ+ are isomorphic to R.
Proof. The correspondence of Corollary 3 clearly exhausts R.
REFERENCE
1. R. Baer, Abelian groups without elemeμts of finite order. Duke Math. J. 3 (1937),68-122.
U N I V E R S I T Y O F WASHINGTON
RECURRENCE TIMES FOR THE EHRENFEST MODEL
RICHARD BELLMAN AND T H E O D O R E HARRIS
1. Introduction and summary. In 1907, P. and T. Ehrenfest [ l ] used a simple
urn scheme as a pedagogic device to elucidate some apparent paradoxes in thermo-
dynamic theory. Their model undergoes fluctuations intuitively related to fluctu-
ations about equilibrium of certain thermodynamic systems. In view of an apparent
discord among physicists [6, pp. 139-145] we shall not try to force an analogy
with entropy.
The original Ehrenfest scheme was defined as follows. Initially, 2/V balls are
divided in an arbitrary manner between two urns, 1 and 2, the balls being numbered
from 1 to 2N An integer between 1 and 27V is selected at random, each such
integer having probability (2/V)~1, and the ball with the number selected is trans-
ferred from one urn to the other. The process is repeated any number of times. If
πγ and n2 are the numbers of balls in urns 1 and 2 respectively before a transfer,
it is clear that the probability is nχ/(2N) that the transfer is from urn 1 to urn 2
and n2/(2N) that it is in the contrary direction.
Let x' (n) be the number of balls in urn 1 after n transfers, and let L 'j9k be
be the smallest integer m such that x' (m) = k9 given that x' (0) == /. If k = /, we
call L'k9k the recurrence time for the state k. If k ψ j , we call L 'j^ the first*
passage time from j to k The distribution of %'(n), known classically, was
derived by Kac [5] as an example of the use of matrix methods. Kac then found
the mean and variance of £/,&> attributing some of his methods to Uhlenbeck
Friedman [4] found the moment-generating function for x1 (n) (for the Ehrenfest
and more general models) by solving a difference-differential equation.
Instead of the original Ehrenfest model, we shall discuss a modified scheme
with a continuous time parameter, which was apparently first suggested by
A. J.F.Siegert [ 9 ] . In this scheme there are two urns and 2N balls initially
divided between them arbitrarily. Each ball acts, independently of all the others,
as follows: there is a probability of (1/2) it + o(dt) that the ball changes urns be-
tween t and t + it, and a probability of 1 — [(1/2)it + o(it)] that the ball remains
Received August 5, 1950, and, in revised form, November 8, 1950.Pacific J. Math. 1 (1951), 179-193.
179
1 8 0 RICHARD BELLMAN AND THEODORE HARRIS
in place between t and t + dt. Standard reasoning then shows that the total proba-
bility of a change by some ball between t and t + dt is Ndt + o(dt), and that
consequently the probability density for the length of time between transfers is
Ne l dt. When a transfer occurs, it is readily seen that the probabilities that it
is from urn 1 to urn 2 or from urn 2 to urn 1, respectively, depend on the relative
number of balls in the two urns exactly as for the original Ehrenfest model. Thus
we see that the present scheme is essentially the original Ehrenfest scheme where
the drawings are made at random times. As we shall see, the time-continuous
scheme is easier to handle analytically.
Let x{t) be the number of balls in urn 1 at time t we shall sometimes speak of
this number as the state of the system. Then x(t) is a random function which can
take integer values from 0 to 2/V; x(t) executes a random walk—with a "restoring
force7-—about the equilibrium value N. It is clear that the random walk is a Markov
process.
Let Lj9k, j f1 k, be the first-passage time from state / to state k that is, Lj9k
is the' infimum of t such that x{t) — k, given that x(0) = /• Let L^^ be the re-
currence time for the state k; that is, L^^ is the infimum of t such that x(t) = k
and x(r) φ h for 0 < r < ί, given that x(0) = k. We shall discuss the probability
distributions of L; jς and L^^
The probability distribution of Lj^ depends, of course, on the size of the
model (that is, on the number N). When it is necessary to emphasize this de-
pendence we shall sometimes employ the notation L- £ in place of £/,&•
We shall use the notation P{A) for the probability of the event A P(A \ B) for
the conditional probability of A, given B E(X) for the mean, or expected value, of
the random variable X. By the distribution of a random variable X we mean the
function (of say u) given by P(X < u). The statement that a sequence of dis-
tributions converges to a distribution F(u) will mean convergence at all continuity
points of F(u).
There are two limiting situations in which the distribution of £/,£ is of interest.
(a) Consider a simple thermodynamic system such as an ideal gas in a con-
tainer. Let us think of the container as consisting of two halves which, however,
are not separated by a partition. Suppose that initially the molecules are spread
in a rather uniform manner through the two halves of the container. According to
classical kinetic theory, if we wait long enough, a time will come, in general,
when all the molecules are in one half of the container. Such events, where the
fraction of molecules in one half of the container is appreciably different from
RECURRENCE TIMES FOR THE EHRENFEST MODEL 1 8 1
(1/2), are evidently enormously rare if the number of molecules is large. Corre-
spondingly, we should like to show that the random variable £#,& > where | k ~~ N \
is of the order of magnitude of N, is very large with high probability when N is
large. Now the mean of L#tk 1S extremely large when N is large. However, as Kac
has observed, the standard deviation is of the same order of magnitude as the
mean. Thus we cannot conclude from the values of the first two moments that L # ^
is large with high probability. We shall show, however, that the distribution of
Ltf 9yE{L^fj£) converges to 1 ~~ e u as N—> °° provided k/N remains less than
some fixed number λ t < 1 (Theorem 1).
The situation with respect to L/ς £, where again k/N < λj < 1, is somewhat
different. If k/N is appreciably different from 0, a very short recurrence time is
not improbable. The distribution of L^^/EiL^^) has for large N a "lump"of
probability of magnitude k/N concentrated near 0, the remainder of the distribution
being exponential (Theorem 2).
(b) In the theory of the Brownian motion and elsewhere in physics and sta-
tistics an important role is played by the stationary Gaussian Markov process z(t)
which we scale so that
E[z(t)] = 0, E[z{t)Y = 1/2.
This process is defined bythe requirement that the joint distribution of z(tι), ,
z(tm) for any distinct numbers t\, , £m is Gaussian and dependent only on the
differences t( ~~tj and that the autocorrelation function is given by
If N is large, the z{t) process, under the conditional hypothesis that z(0) has an
appropriate value, is approximated by the process
x(t) -N
in a sense described in Section 6. (It should be remembered that x(t) depends on
N.) By considering the distribution of Lpt^ where (k — N)/NΪ/2 —» — ξQ < 0,
/Y —> oo We obtain in Theorem 3 the Laplace transform of the distribution of L,
the first time at which z(t) = — ξOy given z(0) = 0. This result is not new, having
been obtained by Siegert [lOj and by Darling (unpublished). However, the present
method of derivation seems instructive.
Results similar to those given under (a) and (b) are obtained for the random
1 8 2 RICHARD BELLMAN AND THEODORE HARRIS
variable L%9k , the first time | x(t) — TV | = N ~~ k9 given #(0) = N,
2 The mathematical model* Suppose that there are initially j balls in urn land
2/V — / in urn 2. Associate with the ith ball a random function x((t) defined as
follows: Xi(t) is 1 if the ith ball is in urn 1 at time ί, and 0 otherwise. From the
elementary theory of Markov processes (see, for example, Kolmogorov [7] ) , we
have
2
We may define the generating function of *, (i) by
P[*i(t)=0] +-sP[xi(t)=l]
Then the generating function of xjit) is, from (1),
or
according as *j(0) is 0 or 1. Since the quantities Λ, (ί), i = 1, * , 2/V, are inde-
pendent, the generating function for x(t) — Σ Xi(t) is2N
(2) Σ Pί*(t) = k\x(0) = j]sk
k=o
= 2"2Λf[l ~ e-f + (1 + e-*)sV[l + e'* + (l - β" ' )* ] 2 *^
= ΣQj,kU)sk,k = 0
where we have introduced the notation Qjfk(t) for P [x(t) = A; | #(0) = y] . Formula
(2) was given by Siegert [9 111
Because of the simple nature of the process under consideration it is easy to
show that Lj9k and L^^ are (measurable) random variables with absolutely con-
tinuous distributions. We omit the proof. We let Py^(α) be the probability density
RECURRENCE TIMES FOR THE EHRENFEST MODEL 1 8 3
Lj,k>
S0
UPj,k(y)dy=P[Ljfk<u].
Define
(5) mk = l/[(N-k)Qk], k<N.
It is convenient to notice that, as N—> °°,
where we have put λ •= k/N, and OiX/N) is independent of λ .
The quantities Ljfk> (?ifc> a n <^ s o o n> depend on the size of the model; when it
is necessary to emphasize this dependence we shall write L-^fQ^ , and so on.
3 Distribution of £/,&, / ψ k. In this section we consider the distribution of
the first-passage time from state / to state k for large N, where | / — k \ is of the
order of magnitude of /V. As far as the limiting distributions are concerned, we can
restrict ourselves without loss of generality to consideration of LΛ',^> k < N. For
example, if j > N > k then we can write
Lj,k ~Lj,N +£tf,fe
The first-passage time from j to N, representing movement toward equilibrium, is
negligible relative to Lsfk a n <^ d ° e s n o t affect the asymptotic result. On the other
hand if N > / > A, we have
and it is not difficult to show that Ljqj is negligible compared with Ln9fg .
If the first passage to the state k occurs at time r , the probability that the
state at time t is again k is Qk,k^ "" T ) . We have therefore
(7) Qjlk(t) = S*Pj,k (r)Qk,k (t-r) dτ, j φk.
Formula (7) is the continuous counterpart of a formula long used for discrete
184 RICHARD BELLMAN AND THEODORE HARRIS
processes and recently exploited by Feller [2] . Taking Laplace transforms of
both sides of (7) we have
S0Qj.k()( 0 e'σt dt = , R(σ) > 0
Γ
Since the quantities Qjfk^ a r e polynomials in e **, as we observe from (2), both
the numerator and the denominator in (8) have a simple pole at σ — 0, and their
quotient is therefore analytic in the circle \σ\ < 1.
For simplicity denote LJ^ ^ by LJj. We have the following result.
THEOREM 1. The distribution function of L^ /m^ ' converges to 1 — e ",
\L > 0, as N—> °°, provided k/N < λ t < 1, the convergence being uniform in k
and u,
The proof will bring out the fact that likewise
(9) E[L[N)]/m^ —» 1 , N—>&, k/N < A x .
Theorem 1 will follow from this lemma:
LEMMA 1. For the complex variable σ, let
Further, let \k(N)] be a sequence of nonnegative integers such that k(N)/N —» λ0
< Xι < 1 as N—> °°. Then (the convergence being bounded and uniform provided
|cr| <σ0 < 1),
(10) lmΦiyi(σ)=- , H < 1 .
Proof of Theorem 1. The function φ^. '( — σ) is the moment-generating
function" of the quantity L^, ymj. . Lemma 1 then implies, as is well known, that
RECURRENCE TIMES FOR THE EHRENFEST MODEL 185
uniformly for u > 0 provided k(N)/N —» λ 0 . Lemma 1 also implies, since we have
convergence in a complex neighborhood of σ = 0, that
so that (11) is still true if we replace m j ^ by E J ^ )
Now if Theorem 1 were not true then an € > 0 and a sequence \h(N)\ ,h(N)/N
< λ t , would exist such that for infinitely many integers N we would have
Extracting a convergent subsequence from \h{N)/N], we are led to a contradiction
of (11).
Proof of Lemma 1. The proof of Lemma 1, which is somewhat indirect, pro-
ceeds as follows. We can obtain an expression for φ\ {&) by substituting cr/m\
for σ in (8), obtaining
(12) φίM) (σ) =T
-j
We can obtain an asymptotic estimate of J γ as we shall see later. However, a di-
rect estimate of J2 appears difficult to obtain. We shall therefore resort to another
expression for φ^ (σ) which is easier to estimate. Having estimates for φ^ '(σ)
and for / ι , we can get an estimate of / 2 , which will be necessary for Theorem 2.
Since a direct proof of Lemma 1 is easy if all terms in the sequence {k(N)\ are
0 we can suppose k > 0. If 0 < k < N we have, from elementary reasoning, the
important relation
On account of the Markovian nature of the process, Ltf^ and L^> 0
a Γ e independent
random variables and the Laplace transform of the distribution of their sum is the
product of the Laplace transforms of their individual distributions. Therefore,
using (8) and (13), we have
(14) E(e~sLN>° ) =E(e~slN>k ) £ ( e ~ s L M ) ,
1 8 6 RICHARD BELLMAN AND THEODORE HARRIS
OΓ
(15) φ[NHσ)= Jβ
βflr,* ( 0 .-**/•**
/o°° PΛ.O (t) Γ*" * dt j Γ ft,, (t) e-'/-* *
Γ Pfe,o (t) e-σt/ * (ft Γ ft,, (t) e-'/ * dt
The advantage of (15) over (12) is that Qk,o(t) is a simpler function than
The numerator of the last fraction in (15) is (l/2)β [N + 1, (l/2)σ/m^ 3 . The
denominator, with the substitution e t = y , becomes
^XOj X "" I ( 1 """" Y ) ( 1 "f" V ) y>CV»'*R' *• fj y ^
We now have to estimate I as N —> °° under the hypothesis k/N—> λ 0 < 1.
[We shall write simply k for k(N).] We shall restrict σ to the circumference of a
circle, say \σ\ = (1/2), since it is clearly sufficient to prove Lemma JL for such
a circle. Write
1 = f6 + f1 =I i + I 2 , 0 < € < l - λ o < l .
Making use of (6) and the fact that (1 — y)k (1 + y)2 N "k increases to a maximum
at y = 1 — k/N and then decreases, 1 — k/N being larger than € for sufficiently
large N9 we have
/T7\ T Z l/iV2 , /•€
L-y)fe(i+:
RECURRENCE TIMES FOR THE EHRENFEST MODEL 1 8 7
= ~ [l + o(l)] +O[(l-e) f e (l +e)2J™ logJV]σ
= — [l+o(l)j +o(mk),σ
where o( ) is independent of σ for \σ\ = 1/2.
To estimate I2 we distinguish the cases \ 0 > 0 and λ 0 = 0. If λ 0 > 0, then
12 can be estimated using the method of Laplace; see [8, p. 77] . We obtain then,
setting k/N = λ, (see (6)),
(18)ττλ(2 ~ λ) 1/2
ΛT(l-λ)
Going back to (15), we obtain (10) from (17) and (18), since
(1/2)B[N + 1, (l/2)σ/mk] = (**/er)[l + <
This completes the proof of Lemma 1 for the case λ 0 > 0. If λ 0 = 0, the integral
12 can be estimated by making a change of the variable of integration which shows
the integral to be asymptotically equivalent to a Beta function. We need not enter
into details.
4. Distribution of Lk9k We shall establish the following result.
THEOREM 2. Assume λ = k/N < λt < 1, and put
Fλ(u) = λ + (1 - λ) [1 - e~( h λ ) u ] , u > 0 .
Then for every b > 0 we have
lim ^ K4^O^- f λ( u ) l = 0
uniformly in k.
Proof. As in the case of Theorem 1, it is sufficient to prove that the Laplace
transform of the distribution of NQ£ L5 ? approaches
λ0 +(l"-λo)V( o)
provided k/N = λ —> λ 0 < 1. (We know from the general theory of Markov proc-
esses that
188 RICHARD BELLMAN AND THEODORE HARRIS
see Feller, [3, p. 325].)
The relation which replaces (7) when / = k is
(19) Qk,k (t)=e'Nt + JΓ* Pk>k (r)Qkιk [t -r) dr ,
the term e t in (19) being the probability that the system remains in state k the
entire time from 0 to t. From (19), we have
(20) / ^We
If we equate the right side of (8), with j — N and with σ replaced by crNQk, to
the rignt side of (15) with σ/τn^ replaced by σNQk, we obtain
or
(21) JΓ Qk.k(t)
To estimate /3 , which is the numerator of (12) with σ replaced by σ*/(l — λ), we
need two lemmas.
LEMMA 2. Given e > 0, let
t: max
RECURRENCE TIMES FOR THE EHRENFEST MODEL 1 8 9
Then
(22) tff(e)=O(logJV), N—>™.
Proof. By (2), QNtΓ(t) is the coefficient of sΓ in
where z — z(t) = 2(1 + e~2t)/(l — e~2t). Since for large N the root of the
equation
(1 - e~2t)N = 1 - δ , 0 < δ < 1 f δ fixed ,
is approximately t ~ (1/2) log Nf it suffices to prove Lemma 2 for the quantities
\cr ( t) — cr \ Itjy(e) = sup ' t : max jj^\ > er ,
- r J
where we have set
(23) ( 1 + 2 S + s 2 ) " :
C C ( ) 2 Q
Suppose e > 0 is given.Choose an arbitrary OC > 1. Let 6ι < e be a positive
number and define
(24) e f f + 1 = e w ( l + W α ) , N = l,2, " .
Note that { βff \ is a bounded increasing sequence. We select €x small enough so
that eN < € , for all N. Now define a sequence ί\ < T2 < as follows:
Fx = tι(βι) ί^+1 for yV > 1 is the maximum of TN and the positive root of
(25) (l/3)[Z(t) -2](l+eH)/eκ = 1/N«.
[Note that zit) is monotone decreasing.] It is then clear from (25) that
(26) tN - (l/2)α log N, N—>oo.
We now wish to show inductively that
icΓ
wω-cW|(27) ^ j <eN f o r t>tN, N = l , 2 , ' .
1 9 0 RICHARD BELLMAN AND THEODORE HARRIS
Clearly (27) holds for N = 1, since βι < e and7 t = t\ Suppose that (27) is true
for a general N From (23), we have
(28) c<»+1> (0 = e<*> (t) + zc\i\{t) + c ί 5 ( 0 ,
c^=cW+2c\l\+c[ί\.
Using (27), (28), and the fact that < % / < > / * + l ) < 1/3, we have for t > ΊH ,
From the definition of €# it is then clear that (27) holds with N replaced by N + 1.
Then t > 7/v implies that the left side of (27) is less than € . Use of (26) now com-
pletes the proof of Lemma 2.
LEMMA 3. Assume k/N < λι. Then
Q J 5 (0 < exp[ ~3(1 - λi)2iV/5] .β
Proof. Lemma 3 is an immediate consequence of a result of S. Bernstein on
sums of independent random variables; see Uspensky [12, p. 205] . To apply
Bernstein's result, we consider the 2N balls as consisting of N pairs, each pair
having initially one ball in urn 1 and one in urn 2, letting Uspensky's random vari-
variable x t be the number of balls from the iih pair in urn 1, minus 1, at time L
Now
N
-p Σ * * = * - * <p\Σ*i<k-N\,
and the applicability of Bernstein's result is obvious.
We now return to the proof of Theorem 2. To estimate the integral /3 defined
in (21), write
( 3 0 ) h= S0
C°(QN,k(t)-Qk)e-σNQktdt +
Write the integral on the right side of (30) as
RECURRENCE TIMES FOR THE EHRENFEST MODEL 191
for an arbitrary € > 0, where ίjy(e) is defined in Lemma 2. Using Lemmas 2 and
3, we have
(31)
\Γ3 I =0{ίog N exp[-3(l - λtfN/5]} .
Thus /3 ~ l/uVcr) Putting this estimate in (21) and recalling from Theorem 1 that
i+°/(i-λ0) '
we get the desired result from (20).
5 Intuitive interpretation. Theorem 1 means intuitively that if we take m^
as our time unit, the attainment of the state k is an occurrence of the "chance"
type; that is, the probability of attaining k during a given time interval is almost
independent of the past history of the process. This interpretation suggests that
Theorem 1 should be true for more general types of processes with a central
tendency.
Theorem 2 seems to mean that if the initial state is k there is a probability λ
of returning J o k before leaving its immediate neighborhood; there is a probability
1 — λ of getting completely away from the neighborhood before the first return; in
this case the first return has the distribution of first passage times given in Theo-
rem 1.
6* Application to stationary Gaussian Markov processes* In Theorems 1 and
2 we considered rare or microscopic fluctuations of x(t). But if N is large x(t) will
for the most part deviate little from its mean value /V, and to consider the ordinary
fluctuations of x(t) we consider
Let ί ι , , t m b e a fixed set of nonnegative numbers. The joint distribution of
Ztf(tι)* * * * 9 Ztfίtm)* given 2^(0) = 0, approaches, as N—* °°, the joint distrib-
ution of z(tι)f , z(tm), given z(0) = 0, where z(t) is the stationary Gaussian
Markov process with
£ [ Z ( t ) ] = 0 , E[z(s) z(s + t)] = (1/2) e " " 1 .
192 RICHARD BELLMAN AND THEODORE HARRIS
Define the random variable L to be the smallest value of t for which z(t) =
~~£o ^ 0> given z(0) = 0. It is intuitively clear that the distribution of L is given
by the limiting distribution of L^9k a s N—* °° provided we let
(32) (fc-jvj^i/a—* _ £ , .
A rigorous proof of this statement is not difficult but we omit it.
To find the limiting Laplace transform for the distribution of Ljv,& u nder the
hypothesis (32), we consider (15) with σ > 0 in place of σ/m^ , and let k =
N — ξN^2 . The substitution e t — y/N^2 puts the denominator in the form
\
7Nl/2j
where Ct is an arbitrary number between 0 and 1/6 . If 0 < y < /Vα, then
Hence,
N1/2I
mξuιn
+ θ(ΛΓ 1 / 2 + 3 α ) ] .
yσ l dy.
The second integral inside the bracket in (33) goes to 0 as /V —> °° .The numerator of (15), with σ in place of σ/m&, is
We thus have the following result.
THEOREM 3. The Laplace transform of the distribution of L is given by
(l/2)Γ(σ/2)(34)
Formula (34) was obtained by Siegert and by Darling through direct consider-
ation of the z(t) process. It is interesting to notice that the present procedure
utilizes (13) which has no counterpart for the z(t) process.
RECURRENCE TIMES FOR THE EHRENFEST MODEL 193
7. Two-sided limits. Let L%\k9 N > k, be the first time | x (ί) - N | = N ~ k9
given x(0) = N. Let L* be the first time \z(t)\ = ξQ > 0, given z(0) = 0. Argu-
ments similar to those used for Theorems 1 and 3 give the following two results,
THEOREM la. Under the conditions of Theorem 1 the limiting distribution of
L%9k/mk is 1 ~~ e ~ 2 u , a > 0.
THEOREM 3a. The distribution of L has the Laplace transform
(l/2)Γ(σ/2)
f e y γσ ι cosh (2ξ0y)dy
8. Added in proof An argument has been found which rigorizes the remarks of
Section 5 and gives a proof of Theorems 1 and 2 for more general processes.
REFERENCES
1. P. and T. Ehrenfest, JJber zwei bekannte Einwande gegen das Boltzmannsche H-Theorem, Phys. Z. 8 (1907), 311.
2. W. Feller, Fluctuation theory of recurrent events, Trans. Amer. Math. Soc. 67 (1949),98-119.
3. , Introduction to Probability, Wifey, New York, 1950.
4. Bernard Friedman, A simple urn model, Comm. Pure Appl. Math. 11 (1949), 59.
5. Mark Kac, Random walk and the theory of Brownian motion, Amer. Math.- Monthly 54(1947), 369.
6. A. I. Khinchin, Statistical Mechanics, Dover, New York, 1949, pp. 139-145.
7. A. Kolmogorov, Analytischen methoden in der Wahrscheinlichkeitsrechnung, Math.Ann. 104 (1931), 432.
8. G. Polya and G. Szegό', Aufgaben u, Lehrs'άtze I, Dover, New York, 1945.
9. A. J. F. Siegert, Note on the Ehrenfest problem, Los Alamos Scientific Laboratory,MDDC-1406 (LADC-438).
10. , On the first passage time problem (abstract)t Physical Rev. 70 (1946),449.
11. , On the approach to statistical equilibrium, Physical Rev. 76 (1949),1708-1714.
12. J. V. Uspensky, Introduction to Mathematical Probability, McGraw-Hill, New York,1937.
STANFORD UNIVERSITY
THE RAND CORPORATION
ON THE APPROXIMATION OF A FUNCTION OF SEVERALVARIABLES BY THE SUM OF FUNCTIONS
OF FEWER VARIABLES
S. P . D I L I B E R T O AND E . G . S T R A U S
1. The problems. Let R denote the unit square 0<x<lf0<y<l9 and
C R the space of all continuous real-valued functions z defined on /?, with norm
11 z 11 defined by | | z | | = ma.x(X9y)€R \z\. Let Ix and Iy denote respectively
the unit intervals 0 < x < 1 and 0 < y < 1; and let Cx and Cy denote respective-
ly the classes of all continuous functions on Ix and ly By an obvious identifi-
cation Cx and Cy may be considered as subsets of C R Let C$ denote the subset
of C R composed of all functions z £ CJJ such that z = / -f g where / £ Cx and
g C Cy C$ is closed (under the above norm)
For z £ CR , define the functional^/i [ z ] by
μ[z] — d i s t [z, Cs] ~ inf | | z — u; | | .
The following problem was posed by The RAND Corporation.
Problem (A): Given z £ C/j and e > 0, give a method for evaluating μ[z~\ to
within € . *
Problem (B): Given z £ C/j and 6 > 0, give a method for constructing
functions / £ C^ and g £ Cy such that
It is our purpose in the present note to solve these problems and to establish
certain generalizations.
Received November 7, 1950.
•Actually, this differs somewhat from the problem as formulated by RAND, which was:Given z and 8, give a method for determining whether μ-[z] < δ This is in all probabilityunsolvable when μ[z] = δ, since any computation of μ[z~\ which can be carried out in afinite number of steps will, in general, yield only an approximation.
Pacific /. Math. 1 (1951), 195-210.
195
196 S. P. DILIBERTO AND E.G.STRAUS
2. The role of the minimizing sequence. We shall now define a few terms by
means of which our procedure can be outlined conveniently.
We shall say that two functions z and z in CR are equivalent if z —z £ Cs,
and shall denote the equivalence of z and z by z ~ z . Clearly, z ~ z implies
μlz] = μ[z]According to the definition of μ [ z ] , there exists a sequence of functions
\wι\, W{ C C$1 such that
v>ί\
Let us define Z( — z — wι; then z^ ~ z and ||2:^|| —> μ [z ] . We shall call a
sequence fzj }, z/ C C/? , a minimizing sequence for z if z t ~ z for all i and
Clearly, both of the proposed problems will be solved once one has constructed
a minimizing sequence.*
We shall introduce a "leveling process," which when applied to z and then
iterated will produce a sequence of functions \zι\ with the properties (1) Z( ~ z
and (2) | |z, || > | |z, + i | | for all i. Properties (1) and (2) imply
lim || zi | | = if > μ[z] .i-oo
That M = yU [z ] , that is, that our "leveling sequence" is in fact a minimizing
sequence, is the principal result of this paper.
This will be established by a "pincers" argument to obtain convergence—
μ [ z j is simultaneously approximated from above and below: For each path in the
class of admissible paths L (defined below) we shall define a functional 77j[z],
over Cβ , with the important property 77 [z ] < μ [z ] . Let
S U P I77/ [ z J I = τf[.z~\leL
Then
-rr[z] < μ [ z ] < M .
* Given a sequence of real numbers ai~^at let us call the integer-valued function N(e)of the real variable € , defined for € ^ 0 , a modulus of convergence for the sequence a j , ifi £ N(ε) implies | α i - σ | < c
While a method for constructing a minimizing sequence answers the questions, thefiniteness of the procedure is satisfactory only when one has an estimate for the modulusof convergence. This will be discussed at the end of this paper.
APPROXIMATION OF A FUNCTION OF SEVERAL VARIABLES BY SUMS OF FUNCTIONS 1 9 7
Our proof is accomplished by showing that 77[_z ] = M, thus implying also that
ττ[z] = μ[z] = M.
3 The main theorem. We shall say that a closed polygonal line is permissible
if it lies entirely within the square 0 < x < 1, 0 < y < 1, and if each of its
sides is parallel either to the x- or to the y-axis.
We enumerate the vertices of a permissible line by (xj>yj)9 j — 1,2, ,
where
l - * 2 f e ι y2kJtl~y2ky *fe + 2n = *fe, ϊk + 2n = ϊk \ k = 1, 2, ' .
To each permissible polygonal line I we can associate a functional π^ \_z~\ with
LEMMA 1. If z ~ z, then ^ [ 2 ] = rr^z].
Proof. Let z(x,y) = z(x,y) + gθc) + h(y) then
2n 2n 2n 2π
Σ(-i);>U;,yy) = Σ(-i)^(*j,yy) + Σ(-i)'β(*j) + Σ>=1 j-ί j-l > = 1
But
Σ (-l)yg(*i) = - Σ g(*Λ-i) + Σ «(*2k) = 0 ,
2n
Hence
2n 2n
Σ (-i)''*(*j.yj)= Σ
that is, 77j[>] = 77 z[z].
We remark that these invariants (under equivalence) ^ [ 2 ] form a complete set
of invariants. That is to say: If 77, [z ] = 77, [2] for all permissible lines, then
198 S. P. DILIBERTO AND E. G. STRAUS
z ~ z . In fact the 77, based on rectangles alone form a complete set of invariants.
In order to relate μ with the 77 we prove the following result.
LEMMA 2. The functional μ [ z ] satisfies μ [ z ] > | 77 [ z ] | for all permissi-
ble lines.
Proof. If we had μ [ z ] < 77 [ z ] | then there would exist a function z ~ z
such that I z || < I 77 [ z ] | and hence :
λ 2n
I w J I - \"il*s\ - 2n
•i 2 n
J = l
which is a contradiction.
Problem A will be solved once we establ ish the following theorem.
THEOREM 1. The functional μ\_z~\ satisfies μ [ z ] = s u p | π ^ [ z ] | = 7 7 [ z ] ,
where the sup is taken over all permissible lines.
As a preliminary to the proof of this theorem we introduce the following level-
ing process :
Given z £ CR , we define the sequences of functions zn(x,y), gn(χ)i AΛ(y)
(n — 1, 2, ) by the relations :
Z i =z Z t Z2n ~ Z 2n-1 βn > Z2n+1 Z2n ^ π >
min
It is obvious that zn ~ z(n = 1,2, ) The passage from z2n~ι t o Z2n reduces
liz2Λ-ilι ^y t-ne maximal amount by which it can be reduced through the sub-
traction of a function of x , while the passage from z2n t° Z 2n+i reduces ||z2nll
by the maximal amount by which it can be reduced through the subtraction of a
function of y. Thus, if we let
= \\z
APPROXIMATION OF A FUNCTION OF SEVERAL VARIABLES BY SUMS OF FUNCTIONS 199
then the Mn form a nonincreasing sequence of nonnegative numbers, so that
limn^a)Mn[^ z ] — Aί [ z ] exists. We have the following obvious result.
L E M M A 3 . The f u n c t i o n a l μ [ z ] s a t i s f i e s μ [ z ] < M [ z ] .
Our solution of problem B will be a consequence of the following theorem.
THEOREM 2. The functional μ [ z ] satisfies μ[z~\ = /W [ z ] .
This, incidentally, will establish the fact that the functional Aί[z] is invariant
under equivalence. The direct proof of this fact might prove somewhat cumbersome.
Keeping in mind the results of Lemmas 2 and 3 we see that both Theorems 1
and 2 are consequences of the following result.
THEOREM 3. The functional π f z ] satisfies ττ[_z~\ = Λ f [ z J .
Proof. We shall call a function z horizontally level if
max z(x,y) = — min z(x,y)
for 0 < y < 1, and we shall call it vertically level if
max z{x,y) = — min z{x,y)
for 0 < x < l.For the sake of brevity we shall use the symbol M instead of Λ/[zJ.
There exists a number N such that M2N < M + €, where e is a small positive
number which is to be further determined later. We now perform the next 2τz steps
of the leveling process on the function z2N
There exists a point (xι9yχ) such that
and since z2N+2n i s vertically level there exists a point (^2,72) with *2
such that
Hence we have
200 S. P . DILIBERTO AND E. G. STRAUS
and since M2N + 2n-ι < M + € and Xι — x2 this implies
M + e-gN+n(Xl)>M + 8 or g ^ + n ( χ 1 ) < e - δ
- M - e - gN+n(Xl) <-M-δ or gN+n(Xl) > δ - e
We therefore have certainly
~ e < g/v + n (^ i ) < e
Thus
and since z2jv + 2rc-i ^ s n o r i z o n t ; a H y level there exis ts a point ( # 3 , 7 3 ) with
} s = 72 s u c n that
By the same process a s we applied to gjV+rc(#i) w e c a n now show that
_1(y2) <2e
hence
fur ther, b e c a u s e z2iV + 2rc-2 i s v e r t i c a l l y l e v e l , t h e r e e x i s t s a p o i n t (χ4,y4) w i th
x4 = %3 s u c h t h a t
R e p e a t i n g t h i s p r o c e s s 2n t i m e s w e f ina l ly o b t a i n a s e q u e n c e of p o i n t s ( % i , y i ) ,
' # > ( * 2 Λ + 1 » 7 2 7 1 + l ) j s u c n t n a t ^ 2 / c ~ ^ 2 / c - l ' 7 2 ^ + 1 = Ύ 2 k ( * = 1 , * # f » )
and
APPROXIMATION OF A FUNCTION OF SEVERAL VARIABLES BY SUMS OF FUNCTIONS 2 0 1
z2N(x2k-vy2k-J >M + S-(22n~1 - l )e (fe = l, ,π + 1 ) ,
z2N(x2k y2k) < - M ~ δ + ( 2 2 n " 1 - l ) e (fe = l, , n ) .
We complete the above sequence of points to form a permissible line by adding
the point U 2 n + 2> y2n^2^ w i t n *2rc + 2 = *2rc+l » 72/1 + 2 = 7ι I f w e construct thefunctional 77 associated with this permissible line then we obtain
h M I = \ττι[z2N\\ = —1 2Π + 2
> It + δ - (2 2 *" 1 - l)e — (M + e) .
Since the choice of 6 was independent of n, we can choose e so that (22n~ι)e
= 6x/2 where e% is an arbitrary small positive number. At the same time we can
choose n so large that
M + € €!
n + 1 2 '
Thus we have: For every βγ > 0 there exists a permissible line such that
or, in other words,
τ r [ z ] > M [ 2 ] .
In conjunction with Lemmas 2 and 3, this proves Theorem 3.
4» The discontinuous case. Examining our method of proof we can make the
following observations:
(1) No essential use was made of the continuity of any of the functions
z(x9y)i g(χ) > h(y) involved in the definition of μ [ z ] Specifically we may define
μ*[z] = infg,h supo<;*£ifo<cy<i I z(x9y) - g(x) ~ h(y) \ ,
202 S. P. DILIBERTO AND E. G. STRAUS
where z is an arbitrary (bounded) function defined for 0 < # < I, 0 < y < 1 and
g(x), h(y) are arbitrary functions defined over 0 < x < 1 and 0 < y < 1, re-
spectively. The definition of 77[z] remains valid for discontinuous z , while Λ/[zJ
can be extended to a functional M * [z ] which is defined for discontinuous (bound-
ed) z, simply by replacing all the max and min symbols in the leveling process by
sup and inf symbols respectively. With very minor modifications of the proof of
Theorem 3 we then obtain the following result.
THEOREM 3*. The functional rr[z] satisfies π[z] = μ * [ z ] = M*[z] ,
where (unless we wish to allow infinite values for these functionals) z is an
arbitrary bounded function.
Theorems 3 and 3* yield the following corollary.
COROLLARY. If z is continuous, then
In other words9 the approximation of a continuous z(x,y) cannot be improved by
permitting discontinuous g(x) + h(y).
(2) The functions 77, [z] are continuous functionals in our metric; more spe-
cifically, we have the following result.
LEMMA 4* // \\z — z\\ < e , then | ^
line.
Proof. We have
< e for any permissible
2n
Hence
2 n
2 n j
2n
Σ (-i)'l>(*/.yj) " *(*>.y>:
1 2n
<— Σ2π > = χ
1ϊ II < — 2ne=e
- 2n
APPROXIMATION OF A FUNCTION OF SEVERAL VARIABLES BY SUMS OF FUNCTIONS 203
As a consequence, 77"[z] is itself a continuous functional, as expressed in thefollowing corollary.
COROLLARY.// || z— z I < € , where z and z are arbitrary bounded functions,then
\μ*{z] ~ μ*[ z]\ <e .
It is also easy to show that the functionals Mn [ z ] and M% [ z ] which arise in
the leveling process are continuous in a similar sense.
5. The n-dimensional case. There was nothing in our treatment which de-
manded that z be a function of two variables only, or even that the variables be
numbers. Most generally we can say:
Let S i , S 2 > , Sβ be arbitrary point sets, and let z(s) be a bounded function
defined over the Cartesian product S — Sx X S 2 X * * * X S^. Let 7\, T2, V
Γj be direct subproducts of the Sj such that T{ Sp Tj unless i = /. Let Elf E2,
• , Eι be the projections of S on 7\ , T2, , T^ respectively. We now define
μ[z] = inf/ 1 , . . . f / ι sup ί € S |2 (s) ~ fi(Exs) - ••• ~
where ^ ranges over all functions defined on 71;.Our permissible lines are now replaced by a rather complicated permissible
array of points t{ ι ... j defined as follows :
(a) ί i is an arbitrary point of S
(b) to every point *ί l f...,, m there exist Z points ^ , . . . , ^ 1 ^ί1, %ίm2 » # * '
^ * i i — , i m , j , j = * i i , i m ί t * i J f * β f i = = * i i » 7 5
(d) the number of points in the set is finite
(e) if til9 9im
= ί / 1 , ,/n
t ' i e n m =n(mod2).
(This last condition is not really necessary but it serves to avoid confusion.)
In order to visualize these sets it might be well to consider the case where S
is three-dimensional Euclidean space; that is, Si is the Λ-axis, S2 the y-axis, and
S3 the z-axis. If we take 7i , T2 , Γ3 as the three coordinate planes then the per-
missible point sets consist of the vertices of closed polyhedral surfaces whose
edges are parallel to the coordinate axes. If we take 7\, T2, T3 as the three
coordinate axes then the permissible point sets consist of the vertices of closed
polyhedral surfaces whose edges are parallel to the coordinate planes.
2 0 4 S. P. DILIBERTO AND E. G. STRAUS
To each permissible point set p we now associate the functional
where the summation is extended over all the N different points of the permissible
set. If we generalize the concept of equivalence so that z ~ z whenever
z(s)-z(s) - / , ( £ i s ) + •'• +fι(Eιs) ,
then 77 [ z ] is seen to be invariant under equivalence.
The leveling process consists in the construction of the sequences
according to the following rules:
)
(j =1," , l;n = 0, l , ) .
We can agaiij define the nonincreasing sequence of nonnegative functionals
- S U PseS
and
M[z] = lim «π[z] .n-»oo
All the above lemmas and theorems remain valid under these new definitions;
and the proofs, while more difficult to state, contain essentially no new ideas.
Probably the greatest deviation from the above proofs takes place in the con-
struction of the permissible set through the leveling process in the proof of Theo-
rem 3. We shall therefore describe that process in greater detail.
Choose N so large that Λί#/+1 < M + €, and let Mtf/ + Λ /+i = M + 8;
then there exist two points tγ and ί4 in S such that Eγ £χ = E γ tϊt ι , and
APPROXIMATION OF A FUNCTION OF SEVERAL VARIABLES BY SUMS OF FUNCTIONS 205
a n d t h a t t h e r e e x i s t p o i n t s t ί t 2 a n d ^ 1 , 1 , 2 i * 1 $ s u c h t h a t E 2 t χ = E 2 1 1 > 2 E2 t ι $ ι
T h e n e x t s t e p in t h e l e v e l i n g p r o c e s s a d d s t h e p o i n t s £ 1 , 3 , ^1,1,3* ^1,2,39
* i , 1 , 2 , 3 9 a n ^ s o on* Af ter nl + 1 s t e p s w e h a v e t h e s e t tχ,iΪ9 9 im (m — 0 , 1 , ,
nl ij € { l , 2 , , I ] ) , w h e r e
(1 ) zMΊ(t, . ... . ) > M — (
In order to form a permissible point set we have to adjoin additional points so that
condition (b) will be satisfied. Condition (b) is already satisfied for all points
tifil9 fim withm < (n — 1)1. The number of points with (n — l ) Z < m < nl is
/ !, and is therefore independent of n. It is easy to see that by adding a fixed
finite number of points (this number A depends on k and I but not on n) we can ob-
tain a point set which satisfies condition (b). Thus the augmented point set satis-
fies conditions (a), (b), (d). Since no points of the form i,/» , , / m , ; , / O Γ ^i,/i, ,
/>ιm»/were constructed in the leveling process we can satisfy condition (c) by de-
finition. For the nonaugmented point set, condition (c) is an immediate conse-
quence of (1) if € is sufficiently small. The augmented part can be constructed so
that (c) is satisfied. We denote the nonaugmented set by p' , the augmenting set by
p".Thus we have constructed a permissible point set; if we form the corresponding
functional 7Tp [ z ] , then we have
KM I =^
3Σ zwl(*Mi, . iJ
> - (B - A)[M - (2nl+2
B
206 S. P. DILIBERTO AND E. G. STRAUS
24 B -
where B — B(n) is the number of points in the permissible set, so that B(n)
with n. For a suitable choice of β and n we haveoo
Hence Theorem 3 is true in this generalized case.
6. Further discussion of the leveling process. While the leveling process
gave rise to a sequence of functions zn ~ z with l i m ^ ^ | | z n | | = μ[<z]> we were
unable to show the convergence of the functions zn. In fact, we have not yet
proved the existence of a function 2 £ C# with z ~ z and | |z | | = μ [ z ] , nor
did we investigate the rate of convergence of | |£n| | . It is the purpose of this
section to treat the last two questions.
In order to prove the existence of z we prove the equicontinuity of the se-
quence \zn\ and thus insure the existence of a convergent subsequence with a
continuous limit z . To this end we first prove :
LEMMA 5. If f%(x), f2M C C Λ and \\fx - / 2 | | < € , then
max fι(x) ~max /2OO : and
Proof. Since
min f(x) = — max Γ "~
it suffices to prove the first statement. Let
then
and hence
min f \{x) ~~ min /2OO < € .
o<*£l
) > fχ(xι) — € = max f \{x) ~~ β
max f2(x) ^ max f \{x) "" β ,
APPROXIMATION OF A FUNCTION OF SEVERAL VARIABLES BY SUMS OF FUNCTIONS 2 0 7
or
max fι(x) ~~ max / 2 M0<x<\ 0 l
Similarly,
max /2(*) ~ max f \{x) <
We define
n
"* ( ) = V ( )
n
hn(y) = Σ hk(y) ,
so that
= z ~ In "" Ίn
The equicontinuity of {z^l will be the direct consequence of the following
result.
THEOREM 4. If for fixed y and Δy we have
\z(x,y) -z(x,y + Δ y ) | < e /or 0 < ^ < 1 ,
then we have
\hn(y)-hn(y+Ay)\ < e (n = l ,2," ) .
Similarly, if for fixed x and ΔΛ; i^e have
\z(x,y) -z{x+άx,y)\ < e for 0 < y < l ,
then we have
< 6 ( n = l , 2 , " ) .
Proof. It obviously suffices to prove the first part of the theorem.
Let z* = z(x,y) — gn{x); then
\z*(x,y) ~ z*(x,y + Δ y ) | = U ( * f . y ) - z(x f y + Δy) | < e for 0 < x <
208 S. P. DILIBERTO AND E. G. STRAUS
Hence, if in Lemma 5 we let fι(x) — z*(x9y) and f2(x) — z*(x,y + Δy), we
obtain
max [z(x,y) — gn(x)] ~ max [z(x, y + Δy) - gn(x)]0i o ς i
min [z(xty) - gπ(*)] - min [z(xty + Δy) - gn(x)]O l 0<x<,\
< 6
< e
z ~~ gn ~~ hn is vertically level. Hence
in [z{x, y)-gn(
in [z (x, yI
If
2 l
~hn{y + Δ y ) |
max [z(* f y) - gn(x)] ~" max [Z(Λ, y + Δy) - g n(*)]
mm , y) - gn(a:)] ~ min [z(x, y + Δy) - gn(x)] I < 6
The discussion so far has failed to settle the questions of the rate of con-
vergence of | | z π | | and of the convergence of zn. We were able to obtain only
partial answers. At the suggestion of the referee we omit the proofs of most of the
following statements; their sequence will have to indicate our derivation.
LEMMA 6. For n>2we have
i i g j > κ ι ι > ι i i ι ι i i ι
LEMMA 7. We have also
APPROXIMATION OF A FUNCTION OF SEVERAL VARIABLES BY SUNK OF FUNCTIONS 2 0 9
> H l w - J I +(22"+1-2)llgJv+nll
Σ ( 2 2 N ' 2 k + 1 - Dl l g^Hfcl l - Σ ( 2 2 " ~ 2 * + 2 - 1 ) I I
THEOREM 5. The norm \\z\\ satisfies
THEOREM 6. For every € > 0 there is an n0 such that for all n > n0 we have
llgn||<(2+e)W/log2n.COROLLARY. The following relations hold:
l im \\gn\\ = K " II*nII = 0 .
DEFINITION. A function z(x,y) is level if it is both horizontally and verti-
cally level.
THEOREM 7. For every z £ CR there is a Z £ CR such that Z is level9
Z ~ z, and μ[z] = \\z\\ .
Proof. According to Theorem 4, the sequence \zn} has a uniformly convergent
subsequence {zni}.het
I = lim zni .
i-oo
According to Theorem 6 we have
lim || zni + 1 — z π j l = 0
hence
z = lim z n i + i .
Of the two functions zn., -z^ i + 1 one is horizontally level, while the other is
vertically level; hence the common uniform limit is level.
Since zni ~ z, we have Z ~ z and
|| z | | = lim | | z π i | | = lim || zn\\ = μ[z] .i-*co
2 1 0 S P. DILIBERTO AND E. G. STRAUS
We remark that the bound obtained for \\gn || in Theorem 6 .does not seem to be
the best possible. In fact in all the cases we have investigated we obtained
| |gΛ | | < c 2 " " . Such an estimate would of course settle the unsolved question of
the convergence of the sequence {z Λ | .
Another unsettled question is that of the existence of a minimizing function z
equivalent to a discontinuous function z . While Theorems 4-7 remain valid with
minor modifications for discontinuous z, Theorem 4 no longer implies the exist-
ence of a convergent subsequence of {zn}.
U N I V E R S I T Y O F C A L I F O R N I A , B E R K E L E Y
U N I V E R S I T Y OF C A L I F O R N I A , L O S A N G E L E S
CONVOLUTION TRANSFORMS WITH COMPLEX KERNELS
I. I. HIRSCHMAN, JR. AND D. V. WIDDER
1. Introduction* In the present paper we shall consider the inversion of a class
of convolution transforms with kernel G(t) of the form
(1.1)
(1.2)
(- co < t < oo) ,
*(•)= Π " " )es/bk
ak = bk + ic^ (A = 1,2, * * •) being a sequence of complex numbers such that
(1.3) Σk=ι
Σ (ck/bk)2 <oo.
This class of kernels is more extensive than that treated previously by the authors,
see [4] , [ 5 ] , [6] , and [7] however the results obtained here are slightly less
precise than those which it was possible to obtain there. We shall show essentially
that if
(1.4) /(*)= / lG(*-t)c iα(t ) ,
and if xx and x2 are points of continuity of CC(t), then
(1.5) lim /fli-oo * 1
*2 DΠ 1 - -
Here D is the operation of differentiation, and e that of translation through the
Received November 20, 1950.Pacific J. Math. 1 (1951), 211-225.
211
2 1 2 I. I. HIRSCHMAN, JR. AND D. V. WIDDER
distance l/α, so that, for example,
α2 / \ 6χ b2
bt b2
1I rll
aχa2
If we replace equation (1.2) and inequalities (1.3) by the more special relations
" / sΛ(1-6) E(s)= Π 1 - - 7 ,
00
(1.7) l im6fc/fc=Ω>0, Σ ( c * / ί > * ) 2 < 0 0 »
we have in addition the complex inversion formula,
(1.8) lim fχ*2 dx fc f(λw + x)K(w) dw = α(* 2 ) ~ Ot(%i) ,
where
(1.9) K(w) = f™ E(s)e~sw ds
and Cχis a closed rectifiable curve encircling the segment [—ίΩ , iΩ] and lying
in the strip | &w \ < Ω/λ . The inner integral in formula (1.8) is to be taken in
the counterclockwise direction.
As one example we may take
v! x Γ(l/2 + v/2)
2
Γ(l/2 + v/2 - s/2) Γ(l/2 + v/2 + s/2) '
G ( 0 = Γ(l/2+,/2)
2 '
CONVOLUTION TRANSFORMS WΓΓH COMPLEX KERNELS
where Hv > — 1 . If
213
/(«)= Ci*1*then
limR -α>
1 - = α ( χ 2 ) - α-1/2+V/2 +fe
and if Hv > 0, then
lim — — / dx I . f{x + ιkw) |_cos w] dw
See [7] and [β] , and [9] . A second example is
E(s) =π2s ΊTVcos I
2 2 2 2 2 2 2
G(t) = - cos e%(e*)77 2
for - 1 < Rz < 1. If
77
then
lim J» 1
-1/2
D
- v/2 + k1 -
~l/2 + v/2 +kj
See [ 2 ] .
2. Inversion of a class of convolution transforms. We assume as given through-
out this section a sequence, {α }J°, of complex numbers α^ = b^ + j'c^ subject
2 1 4 I. I. HIRSCHMAN, JR. AND D. V. WIDDER
to the restrictions
(2.1) Σ (l/bk)2
k=i
We define the entire functions
(2.2) EΛ,n(s) =
Σ (tk/bk)2 <<».
k=i
= Π (I ~ s/ak)k=m+l
= Π
The definition of Em(s) is significant because
00 / s \
π i--••
00
exp ΣΛ + l -f- i
and because the series Σ ^ + 1 | a^ \ ~2 , Σ ^ + 1 c^/bj^b^ -f ί'c^) converge as a
consequence of (2.1) and Schwarz's inequality. Similarly, Fm(s) is well defined.
We define
(2.3)
We also set
(2.4) βχ (m)
THEOREM 2a.
1
(*=0, l , ) .
β2(m) = m in (bk,6o
, m yjf ± , z,, ) ,
CONVOLUTION TRANSFORMS WITH COMPLEX KERNELS 215
then we have
A.
β.
C
D.
f_m \Gm(t)\e-σt dt <
P«(fl)G0(t)=G«(t);
(d/dt)kGm(t)=O(e^t), *-»+«
= 0,l, ) ,
Conclusion A is an immediate consequence of Hamburger's theorem; see
[4, pp. 141-144]. We define g(u) = e*" 1 for -co < u < l , and ^(w) = 0 for
1 < u < °° , and we set
=α^ sgn bk\exp[ick(t ~ bϊ1
It is immediately verifiable that
for —oo < Rs < bjς if bjς > 0, and for b^ < fts < oo if b^ < 0. Let
g i * g 2 ( O = Xoo gi (ί "" u)g2 (u) si" ,
and so on; then by the convolution theorem for the bilateral Laplace transform
we have
£ ^ g»+i * g«+2 * •'• *gn(t)e-stdt = [ £ . , „ ( « ) ] " *
for βι(m) < Rs < β2(m). From the complex inversion formula for the bilateral
Laplace transform we obtain
*g«+2 * ••• * g » ( t ) =,2πi
2 1 6 I. I. HIRSCHMAN, JR. AND D. V. WIDDER
Since
= 0
for —oo < t < oo , it follows that
* ••• *gn(t) =Gn(t) (-oo< ί <oo).n-oo
See [4; pp. 139-145]. It is easily seen that
SZ kk(t)\e~stdt = [(1 - 8/bkyfo\bk/ak\Tι ,
for -oo < Hs < bk if bk > 0, or for bk < Hs < oo if bk < 0. By Fatou's
lemma we have
f_l K(t)\e-σt dt < lim inf j Γ | g l l + ι * * gn(t) \e~σt dt ,
< lim inf X " |g B + 1 | * ••• * | g π ( ί ) | e " σ t d t
so tnat conclusion B is established.
Conclusion C follows from the identity
P»(D)est = e s t f [ ( 1 -
Conclusion D may be established by shifting the line of integration in the integral
defining Gm(t) to Rs = yx and Hs = γ2 . See [4; pp. 152-154] .
In what follows we shall write G(t) for G0{t).
THEOREM 2b. //
(a) G(t) is defined as in Theorem 2a,
(b) βx(o) < c < /32(o), c + n > A(o), c + γ2 < 02(o),
CONVOLUTION TRANSFORMS WITH COMPLEX KERNELS 217
(c) α(ί) is of bounded variation on every finite interval, d(t) — 0(e7i ) as
t —•> - <» , α(ί) = 0(e7^) as t —> + oo $
(d) flnΦ) is defined as in equation (2.3),
(e) / ω = / ω G ( * - ί ) β
c l d α ω ,
(f) ^i and %2 are points of continuity of 0C(ί),
then
lim = a(xa) - α (
From assumption (c) and from conclusion D of Theorem 2a we may show, using
integration by parts, that each of the integrals
converges uniformly for x in any finite interval. Since Pm(D)G(t) = Gm(t) by con-
clusion C of Theorem 2a, it follows (see [4; pp. 167-170]) that
(2-5) P. &>)/(*)= f_lGΛ(χ-t)ectda(t)
Multiplying by e~cx and integrating by parts, we have
(— 00 < x < OO) ,
α(t)dt
a(t)dt.
Since this integral converges uniformly for Λ; in any finite interval, we obtain
e-c*PΛφ)f(x)dx
Γ2 dx foo I 3
oo I ->„ *>] α ( t ) Λ
218 1. I. HIRSCHMAN, JR. AND D. V. WIDDER
»i α(t )Λ
We thus need only show that if x is a point of continuity of α(ί) we have
(2.6) lim fW Gm(x - t)β-e<*-*> α ( t ) dt = α(x) .
We shall first show that for any 6 > 0 we have
(2.7) lim G«(t)e- C t α (x - t) dt = 0 .
Using assumptions (a) and (b) we see that it is enough to prove that for any
δ with βiiO) < δ < /32(0), we have
0 so small that ^ ( 0 ) < 8 - 2η < 8re
(2.8)
Choose rj >
\t\ > 6 we have
"" (sinh
so that it is enough to prove that
lim /°° \Gn(t)\e~$t [sinh ηt]2dt = 0
)S2(0). For
Using conclusions A and B of Theorem 2a we see that
C \G.W\e-* l,iihηtl'dt
F . ( 8 - 2 τ > ) Em(B)
and equation (2.8) follows from this. We assert that
(2.9) lim £ Gm(t)e'ct dt = 1
CONVOLUTION TRANSFORMS WITH COMPLEX KERNELS 219
(2.10) lim^sup f_l \Ga(t)\e-ctdt =
These results are immediate consequences of conclusions A and B of Theorem 2a.
Now x being fixed and 77 > 0 being given, let us choose e > 0 so small that
I (X(t) — a(x) I <_ 77 for I ί — x I < € . We have
j Γ G Λ s - O ^ ^ α ί O d t - <*(*) =ίi +12 +I3 ,
where
h =
[a(x~-t)-a(χ)]dt
[a.(x-t)-a(x)]dt
We have limm-oo/i = 0 by equation (2.9), limm-»co/2 ~ 0 by equation (2.7), and
lim supm-»oo|/3 j <. 77 by equation (2.10). Since 771s arbitrary our demonstration is
complete.
3. Complex inversion formulas* In this section we restrict our attention to a
much more special class of kernels. We suppose that
(3.1)
We define
(3.2)
(3.3)
bk>0, bk~Ωfc
77 IB—
H(λ,s) =
E(s) = Π
λ2 + (1-λ 2 ) \ak\bk
bl - s2 (0 < λ < 1)
2 2 0 I. I. HIRSCHMAN, JR. AND D. V. WIDDER
The product (3.3) is defined for s f^ h^ {k — 1,2, •) since it can be rewritten
ΠH{λ,s) =
i-^-+(i-λ2)\<*k\ ~ bk
π
and assumption (a) implies that ΣJ° [ | α& | — i^] ^^""ι i s convergent. We define
(3.4) β = m i n 6fe .
THEOREM 3a. //
1 ri«> β^£(λs)
2τri E(s)ds
then
A. G(λ,w) is analytic for \&w\ < Ω(l - λ)
(d/dw)kG(\,w) = 0(e 7iu) (u
(0 < λ< 1) ,
= 0 , 1 , . . . ) ,
where yι > —β, γ2 < β, uniformly for \v\ < Ω ( l — λ ) "~" 6 , e > 0.{Here
w — u + iv.)
• ' - o o<//(λ,σ), - β<σ< β.
We shall write G(ί) for G(0, ί) .
We assert that
CONVOLUTION TRANSFORMS WITH COMPLEX KERNELS 221
(3.5) log \E(σ + i r ) I ~ Ω | τ I (r —> ± oo)
uniformly for σ in any finite interval. We define
00
We have
from which it follows that
E(s) » 4(3.6) lim — = Π Hr
uniformly for 0 < 6 < | arg s\ <π — € . From [ l , pp. 267-279] we have that
log |£* (cr + ir) ~ Ω | τ | as r —> ±°°, uniformly for α in any finite interval.
Relation (3*5) now follows.
Conclusion A follows immediately from (3.5) and the definition of G( h,w).
Conclusion B is a consequence of (3.5) and Hamburger's Theorem. The two con-
clusions C are obtained by shifting the line of integration in the integral defining
G(λ,ί) to Rs = 7i , and to Hs = γ2 , respectively. See [6, pp.688-691]. To
establish conclusion D we introduce the functions
Π i-^Hγ.
Π - ik=i \ akl
It is immediate that
lim G π (λ,ί) = G(λ,t) ( - oo < t < oo) .
222
We define
I. I. HIRSCHMAN, JR. AND D. V. WΓDDER
where j(t) = 0 for -co < t < 0; ;(0) = 1/2; j(t) = 1 for 0 < t < oo.
It is easily verified that for ~~&£ < cr < b^ we have
- λV/α?
Just as in §2 we may show that
G n ( λ , ί ) = lim — [/ii(λ,t) * •••* hn(λ,t) * h π + i ( 0 f t ) * •••* h«(θ f t ) ] .w- oo at
Here h1*h2(t) ~ X°^ Aι(ί ^u)dhι(u). Note that this differs from the convention
employed in §2. By Fatou's lemma,
< lim inf f™ e'σt \dhι(\,t) * *hn(λ,t) *hn+1(O,t) * •• *Λm(θ,t)
< lim inf Π 'C
<Π Π
By Fatou's lemma, again,
<π
< Π λ2 + ( 1 - λ 2 )\ak\bk
CONVOLUTION TRANSFORMS WITH COMPLEX KERNELS 2 2 3
This completes the proof of the theorem.
We define
(3.7) K(w) = f* E{s)e'sw ds .
It follows from relations (3.1) that given € > 0, for all sufficiently large r wehave log | £ * (reiθ) | < (e + | sin θ\ )Ωr. See [l, pp.267-279] . Fromequation
(3.6) it follows that
log \E{reiθ)\ < (e + |sin 0 | ) Ω r
for r sufficiently large. Using this inequality and rotating the line of integration
in the integral defining K{w) we can show that K(w) is analytic and single valued
in the w -plane except on the segment [— iΩ , iΩ] . It may also be shown, see
[l, pp.295-311] , that if C is a closed rectifiable curve encircling [~iΩ , iΩ]
then
(3.8)
the integration proceeding in the counterclockwise direction.
LEMMA 3b. If C\ is a closed rectifiable curve encircling [—ίΩ , iΩ]and
contained in the strip \v\ < Ω/λ , then
fc^G(\w+x-t)K(w)dw=G(\,x-t),
the integration proceeding in the counterclockwise direction.
We have
1
2 771- / G(\w+χ-t)K(w)du
2 2 4 I. I. HIRSCHMAN, JR. AND D. V. WIDDER
E(λs)
2771 E(s)ds
THEOREM 3 C //
(a) G(t) is defined as in Theorem 3a
(b) -β < c < β, -β < c + yι , c + γ2 < β
(c) α(ί) is of bounded variation on every finite interval and
α ( t ) = (βn*) ( t_»+oo), α ( t ) = (ey2<) (t_»-oo)
(d) f(w) = Jl0^ G(«; - ί)e c ί r fα(ί)
(e) Jζ(ιt ) is defined as in equation (3.7)
(f) C\ is defined as in Lemma 3b
(g) %ι and x2 are points of continuity of CX(ί), then
lim f*2 e~cx dx - i - Γ /(λ. + ^ W ώ ^ α f c l - α f e ) .λ - l - x l 277 I ^ λ
It follows from assumption (c) and from conclusion C of Theorem 3a that the
integral defining f(w) converges uniformly for w in any compact set contained in
the strip I AM; I < Ω . Hence
x)K(w)dw2πi
by Lemma 3b. The proof may now be completed exactly in the manner of Theo-
rem 2b.
4. Remark. If it is assumed that the roots of E(s) occur in conjugate pairs,
then equation (1.5) can be established under conditions less restrictive than (1.3).
A discussion of this case is given in the Master's thesis of Mr. A. 0 . Garder [3],
written under the direction of one of us.
CONVOLUTION TRANSFORMS WITH COMPLEX KERNELS 2 2 5
R E F E R E N C E S
1. V. Bernstein, Lemons sur ίes progres recents de la theorie des series de Dirichlet,Paris, 1933.
2. R. P. Boas, Jr., Inversion of a generalized Laplace integral, Proc. Nat. Acad. Sci.U. S. A. 28 (1942), 21-24.
3. A. O. Garder, The inversion of a special class of convolution transforms, Master'sThesis, Washington University, 1950.
4. I. I. Hirschman, Jr. and D. V. Widder, The inversion of a general class of convo-lution transforms, Trans. Amer. Math. Soc. 66 (1949), 135-201.
5. , A representation theory for a general class of convolution transforms,Trans. Amer. Math. Soc. 67 (1949), 69-97.
6. , Generalized inversion formulas for convolution transforms, Duke Math.J. 15 (1948), 659-696.
7. , Generalized inversion formulas for convolution transforms, II, DukeMath. J. 17 (1950), 391-402.
8. H. Pollard, Studies on the Stieltjes .transform, Dissertation, Harvard; Abstract48-3-117, Bull. Amer. Math. Soc. 48 (1942), 214.
9. D. B. Sumner, An inversion formula for the generalized Stieltjes transform, Bull.Amer. Math. Soc. 55 (1949), 174-183.
10. D. V. Widder, The Laplace transform, Princeton University Press, Princeton, 1941.
WASHINGTON UNIVERSITY
HARVARD UNIVERSITY
A THEOREM ON RINGS OF OPERATORS
IRVING K A P L A N S K Y
1. Introduction. The main result (Theorem 1) proved in this paper arose in
connection with investigations on the structure of rings of operators. Because of
its possible independent interest, it is being published separately.
The proof of Theorem 1 is closely modeled on the discussion in Chapter I
of [3] . The connection can be briefly explained as follows. Let N be a factor of
type l i t ; then in addition to the usual topologies on /V, we have the metric defined
by [ [^ l ] ] 2 = T(A*A), T being the trace on N. Now it is a fact that in any
bounded subset of N, the [[ ]]-metric coincides with the strong topology—this is
the substance of Lemma 13.2 of [3] . In the light of this observation, it can be
seen that Theorem 1 is essentially a generalization (to arbitrary rings of operators)
of the ideas in Chapter I of [3] .
Before stating Theorem 1, we collect some definitions for the reader's con-
venience. Let R be the algebra of all bounded operators on a Hubert space // (of
any dimension). In R we have a natural norm and *-operation. A typical neighbor-
hood of 0 for the strong topology in R is given by specifying e > 0, ξ 1, ,
ζn C H, and taking the set of all A in R with | |>4^ j | | < e; for the weak topol-
ogy we specify further vectors Tjί9 , T)n £ // and take the set of all A with
I {A ξi, Ύ)ι)\ < e. By a *-algebra of operators we mean a self-adjoint subalgebra
of/?, that is, one containing A whenever it contains A; unless explicitly stated,
it is not assumed to be closed in any particular topology. For convex subsets of
R, and in particular for subalgebras, strong and weak closure coincide [2,Th 5]
An operator A is self-adjoint if A* — A, normal if A A —A A, unitary if A A*
— A A — the identity operator /.
2. The main result. We shall establish the following result.
THEOREM 1. Let M, N be *-algebras of operators on Hubert space, M C N9
and suppose M is strongly dense in /V. Then the unit sphere of M is strongly dense
in the unit sphere of N.
Received October 6,1950. This paper was written in connection with a research projecton spectral theory, sponsored by the Office of Naval Research.
Pacific J. Math. 1 (1951), 227-232.
227
228 IRVING KAPLANSKY
We shall break up the early part of the proof into a sequence of lemmas.
Lemma 1 is well known and is included only for completeness.
LEMMA 1. In the unit sphere of R> multiplication is strongly continuous, joint-
ly in its variables; and any polynomial in n variables is strongly continuous,
jointly in its arguments.
Proof* It is easy to see that multiplication is strongly continuous separately
in its variables, even in all of R. Consequently [ l , p.49] we need only check
the continuity of AB at A — B — 0. Since \\A\\ < 1, this is a consequence of
\ \ A B ξ \ \ < \\A\\ \ \ B ξ \ \ < \ \ B ξ \ \ .
Since addition and scalar multiplication are continuous (in all of R), the con-
tinuity of polynomials follows.
The precaution taken in the next lemma, in defining the mapping on the pair
{A,A ), is necessary since A —> A is not strongly continuous.
LEMMA 2. Let f(z) be a continuous complex-valued function, defined for
\z\ < 1. Then the mapping (A, A*) —> f(A) is strongly continuous on the
normal operators of the unit sphere of R,
Proof. We are given a normal operator Ao with 11 >4oil S 1> a positive e ,
and vectors ξ( in // (i = 1, , n). We have to show that by taking A, A* to be
normal with norm < 1, and in suitable strong neighborhoods of AQ, AQ, we can
achieve
ω l | [ / u ) - / U o > ] £ . - l l < e .
By the Weierstrass approximation theorem, there exists a polynomial g in two
variables such that
(2) |g(z,z*) " f(z)\ < e / 3 ,
for \z\ < 1, z* denoting the conjugate complex of z. By elementary properties of
the functional calculus for normal operators, we deduce from (2):
(3) \\giAtA*)-fU)\\ < e / 3 ,
(4) \\g(Λo,A*)-f<Ao)\\ < e/3.
By Lemma 1, if we take A, A* in appropriate neighborhoods of AQ, A%, we have
A THEOREM ON RINGS OF OPERATORS 2 2 9
(5) \\[gU,Λ*)- g(AΰtAt)]ξi\\ < e/3.
By combining (3), (4), and (5) we obtain (1).
The next lemma follows from Lemma 2 as soon as it is admitted that * is
strongly continuous on unitary operators. This can, for example, be deduced from
two known facts: (a) the strong and weak topologies coincide on the set of unitary
operators, and (b) * is weakly continuous.
LEMMA 3. Let f be a continuous complex-valued function defined on the
circumference of the unit circle. Then the mapping U —> f(U) is strongly con-
tinuous on the set of unitary operators.
The Cayley transform is the mapping A —> (A — i)(A + i) 1 ; it is defined
for any self-adjoint operator and sends it into a unitary operator.
LEMMA 4. The Cayley transform is strongly continuous on the set of all self-
adjoint operators.
Proof. We have the identity
(6) 04 - i)(A + iΓι - U o - i)(A0 + iΓι = 2i(A + i)'1 (A - AQ)(A0 + i)~ι .
When A is self-adjoint, we have | | (A + i)~ ι | | < 1. In order to make the left side
of (6) small on a vector ξ, it therefore suffices to make A — Ao small on the
vector 04 0 + i) ι ξ.
We shall prove a stronger form of Lemma 5 below (Corollary to Theorem 2).
LEMMA 5. Let h be a real-valued function defined on the real line, and sup-
pose that h is continuous and vanishes at infinity. Then the mapping A —* h(A)
is strongly continuous on the set of all self-adjoint operators.
Proof. Define
f{z) = h[-i(z + 1)U ~ 1 Γ 1 ] for \z\ = 1 , z φ\%
=^0 for z = 1.
Then f is continuous on the circumference of the unit circle. Moreover,
h(A) = f[(A - i)U + i ) " 1 ] .
The mapping A —* h(A) is thus the composite of two maps: the Cayley transform,
2 3 0 IRVING KAPLANSKY
and the mapping on unitary operators given by /. By Lemmas 4 and 3, these latter
two maps are strongly continuous. Hence so is A —» h(A).
Proof of Theorem 1. There is clearly no loss of generality in assuming M and
N to be uniformly closed, for the unit sphere of M is even uniformly dense in the
unit sphere of its uniform closure.
Let us write Z for the set of self-adjoint elements in M, and Zγ for the unit
sphere of Z. Let B be a given self-adjoint element in N, | | B | | < 1. By hypothe-
sis, B is in the strong closure of M. We shall argue in two successive steps that
B is actually in the strong closure of Z\ We begin by remarking that B is in the
weak closure of M, since the latter coincides with the strong closure of M Now *
is weakly continuous, and hence so is the mapping A —> (A + A*)/2 This
mapping leaves B fixed, and sends M onto Z; hence B is in the weak closure of
Z. Since Z is convex, this coincides with the strong closure of Z.
Let h(t) be any real-valued function of the real, variable t which is continuous
and vanishes at infinity, satisfies | h (t) \ < 1 for all t, and satisfies h(t) —t
for I ί I < 1. We have that h(B) — B. Also h can be meaningfully applied within Z,
since we have assumed M to be uniformly closed, and in fact h(Z) — Zχ> By
Lemma 5, the mapping A —* h (A) is strongly continuous on self-adjoint oper-
ators. Hence B is in the strong closure of Z ι .
This accomplishes our objective as far as self-adjoint operators are concerned.
To make the transition to an arbitrary operator, we adopt the device of passing to
a matrix algebra.1 Let N2 be the algebra of two-by-two matrices over N. In a
natural way, TV 2 is again a uniformly closed *-algebra of operators on a suitable
Hubert space (compare §2.4 of [3] )• It contains in a natural way M2, the two-by-
two matrix algebra over M. The strong topology on N2 is simply the Cartesian
product of the strong topology for the four replicas of N; thus M2 is again strongly
dense in N2 Now let C be any operator in N, \\C\\ < 1. We form
D -
and we note that D £ /V2> 0 * ~ D9 WOW < 1. Let U be any proposed strong
neighborhood of D. By what we have proved above, there ex i s t s in U a self-adjoint
element F ,
1 1 am indebted to P. R. Halmos for this device, which considerably shortened rayoriginal proof of Theorem 1.
A THEOREM ON RINGS OF OPERATORS 231
_ /G ΛF" V *)
with F G M29 | | F II < 1. By suitable choice of U we can make H lie in a given
strong neighborhood of C. Also | | F | | < 1 implies \\H\\ < 1. This proves that C
lies in the strong closure of the unit sphere of M, and concludes the proof of
Theorem 1.
3 Remarks, (a) Since strong and weak closure coincide for convex sets, we
can, in the statement of Theorem 1, replace "strongly" by "weakly" at will.
(b) From Theorem 1 we can deduce that portion of [2, Th.8] that asserts
that a *-algebra of operators is strongly closed if its unit sphere is strongly
closed; but it does not appear to be possible to reverse the reasoning.
(c) As Dixmier has remarked [2, p. 399], Theorem 1 fails if M is merely
assumed to be a subspace (instead of a *-subalgebra).
4. Another result. In concluding the paper we shall return to Lemma 5 and
show that the hypothesis can be weakened to the assumption that h is bounded
and continuous. It should be noted that we cannot drop the word "bounded," since
for example it is known that the mapping A —> A is not strongly continuous.
Actually we shall prove a still more general result, which may be regarded
as a generalization of Lemma 4.2.1 of [3]
THEOREM 2. Let hit) be a bounded real-valued Baire function of the real
variable tf and Ao a self-adjoint operator. Let S be the spectrum of AOf and T the
closure of the set of points at which h is discontinuous; suppose S and T are
disjoint. Then the mapping on self-adjoint operators, defined by A —> h(A)9 is
continuous at A = Ao.
Proof. We may suppose that
(7) |A(ί) | < 1
for all t. Given 6 > 0, and vectors ξ^% we have to show that for A in a suitable
strong neighborhood of Ao , we have
(8) \\ίh(A)-h(A0)]ξi\\ <e.
Choose a function k (t) which satisfies: (a) k is continuous and vanishes at in-
finity, (b) k(t) = 1 for t in S9 (c) k(t) = 0 for t in an open set containing Γ. Define
232 IRVING KAPLANSKY
p — hkf q — 1 ~~ k ~f" hk* Then p = q = A on S, and so
(9)
Also p and ςr — 1 are continuous and vanish at infinity. Hence Lemma 5 is appli-
cable, and for a certain strong neighborhood of Ao we have
(10) II [p(A)-p(A0)]ξi || <e/4, || [q(A) - q(A0)] ξt \\ < e/2 .
The following is an identity:
(11) h = (l-h)p + hq.
From (9) and (11) we get
(12) h{A) ~h(A0)= [1 - A M ) ] [ P U ) - p U o ) ]
From (7), (10), and (12), we deduce (8), as desired.
If in particular h is continuous, then Γis void and we get a simplified corollary.
COROLLARY. Let h(t) be a continuous bounded real-valued function of the
real variable t. Then the mapping A —> h(A) is strongly continuous on the set
of all self-adjoint operators.
REFERENCES
1. N. Bourbaki, Elements de Mathe'matique, Livre III, Topologie Generate, ActualitesSci. Ind., No.916, Paris, 1942.
2. J. Drxmier, Les fonctionelles lineaires sur I'ensemble des operateurs homes d'unespace de Hubert, Ann. of Math. 51 (1950), 387-408.
3. F . J. Murray and J. von Neumann, On rings of operators /K, Ann. of Math. 44 (1943),716-808.
UNIVERSITY OF CHICAGO
AN ITERATIVE METHOD FOR FINDING CHARACTERISTIC
VECTORS OF A SYMMETRIC MATRIX
W. KARUSH
1. Introduction* Given a real symmetric linear operator A on a vector space 8,
we wish to describe a procedure for finding a "minimum" characteristic vector
of A, that is, a characteristic vector with least characteristic value, supposing
such to exist. The method to be used is, in a general way, the following. Select
an initial vector x° and a positive integer s > 1. Imbed x° in an s-dimensional
linear subspace 8° (appropriately selected). Determine the next approximation xι
as the minimum characteristic vector relative to this subspace (to be defined
later). Next, imbed Λ;1 in an s-dimensional subspace 8 and determine x2 as the
minimum characteristic vector relative to this subspace. Proceeding in this manner,
construct a sequence of subspaces S°, S 1 , of fixed dimension s, with a
corresponding sequence of vectors xι,x2, . It is to be expected that under
appropriate hypotheses the sequence of vectors will converge to a minimum char-
acteristic vector of A.
We shall treat the case when 8 is of finite dimension n, and 8 ι is chosen as
the subspace spanned by the vectors χι, Ax1, A2xι, , As~ιxι We shall es-
tablish the desired convergence under these circumstances, the sequence {x1}
satisfying at the same time a relation xι + i = xι + Ύ]1 with (xι, Tjι) = 0. The main
result is formulated in Theorem 2 of §6. An analogous result holds for a "maximum "
characteristic vector.
It is of interest to compare the present iteration method with what might be
called Rayleigh-Ritz procedures. In the latter, one fills out the space 8 by a
judiciously chosen monotone sequence of subspaces
8! C 8 2 C S 3 C . . . (dim 8; = i)
of increasing dimension. One then obtains successive approximations to a mini-
mum vector of A by determining minimum characteristic vectors of the successive
Received May 20, 1950. The preparation of this paper was sponsored (in part) by theOffice of Naval Research.
Pacific J. Math. 1 (1951), 233-248.
233
2 3 4 W. KARUSH
subspaces. This procedure has the serious computational drawback that to obtain
an improved approximation a problem of increased complexity, that is, of higher
dimension, must be solved. This restriction is important even in the finite dimen-
sional case where the iteration, in theory, terminates in a finite number of steps.
The method of the present paper, however, requires only the solution of a problem
of fixed dimension s at each step, the dimension s being chosen from the outset
as any desired value. The 8* form a chain of subspaces in which successive sub-
spaces 8* and 8 ι + 1 overlap in xi+ι in general this chain will be infinite even
when 8 is finite-dimensional. Thus the method is useful where it is desired to fix
beforehand the degree of complexity for all steps; and yet a great many iterations
may readily be performed. This is the case with high speed computing machines.
The present procedure may be interpreted as a gradient method; cf [ l ] . For
s = 2, in the equation xι+ι = xι + rj1, η* is a multiple of the gradient at x = xι
of the function Gc, Ax)/(x, x) For 5 > 2, the vector r] contains higher order terms.
The applicability of the present procedure with s = 2 to quadratic functionals in
infinite-dimensional spaces has been pointed out to the author by M.R.Hestenes
in conversation, and has been outlined by L. V.Kantorovitch [2] .
2 Subspaces. Before describing in detail the iteration procedure to be used,
and proving its convergence, we find it convenient to formulate some preliminary
results. In this section we construct an orthogonal basis for the space spanned
by the powers of A operating on a fixed vector x; in the next section we describe
the characteristic roots and vectors relative to certain subspaces of this space.
We shall encounter polynomials py(λ) of central importance. In these two sections
we shall be treating, essentially, only one level of the iteration. Accordingly, the
superscript i denoting the various steps of the iteration will not appear until §4,
where we are concerned with the progression from one level to the next.
Let S denote the ^-dimensional space of π-tuples of real numbers; by vector
we understand always an element of 8. We consider a linear operator A on 8 which
is real and symmetric; that is, one for which Ax is a real vector and
(Ax, z) = (x,Az)
for arbitrary real vectors x, z. A characteristic number (root, value) of A is a
number λ for which there exists a non-null vector y such that
Ay = λy .
There are n (real) characteristic numbers (counting multiplicities).
CHARACTERISTIC VECTORS OF A SYMMETRIC MATRIX 2 3 5
With a non-null vector x we associate the number
( . (x,Ax)
(pc,x)
and the vector
ξ{x) =Aχ- μ{x)x .
Let λ m j n ( λ m a x ) be the least (greatest) characteristic root, of A. It is well known
that
(1) λ m i n = min μ(x) , λ m a x = max μ(x) , (x £ £ ) .xέo xέo
For a non-null vector x we define the subspaces
< l j ( * ) = (x,Ax, ~>,A>-ιx) (j = 1,2,3, • • • ) ,
α ( x ) = (x,Ax,A2x, •••) ,
where, in each case, the right side of the equation denotes the space spanned by
the designated vectors. The space U (x) is the smallest invariant subspace con-
taining x denote its dimension by r = r(x) Clearly dι C fl2 C C flr = fl,
where " C " denotes strict inclusion. The space CL contains r independent char-
acteristic vectors of A. We now construct an orthogonal basis for Gy .
LEMMA 1. Let the vectors ξj (j — 0,1, , r) be defined by
(2) £o=x, £i=Aξo- ( ())
ξ Aξj-μjξj- tj
ti = TT^T U = 1,2, ••-, r - 1) .I b l
/or /, A = 0,1, , r ~ 1, zi e Aαve ζj φ 0,
(3) Gj + iGO = ( o , ^ i , * '* , ξj) , (ξj , ξk) = 0 ,
The lemma may be verified directly by induction. We remark that ξΓ — 0.
236 W. KARUSH
LEMMA 2. Let the polynomials py(λ) (/ = 0 , 1 , , r) be defined by
Po(λ)=l, pi(λ) = ( λ - μ 0 ) , PaOO = (λ-μo)(λ-/*i) ~ *? .
PJ + ι(λ)=P J-(λ)(λ-^.)-t/p J..1(λ) 0 = 1,2, — , r - 1 ) .
Suppose B is an invariant subspace containing x; write
(4) x = myi + a 2 y 2 + ••• + aι yι
in terms of a basis of characteristic vectors of B Then
(5) £ ; = a ^ y C λ J y ! + a 2 p ; ( λ 2 ) y 2 + ••• + aιpj(λι)yι ( j = 0,1, •••, r ) ,
where λ k is the characteristic number of yk
The lemma follows immediately from the definitions (2).
The polynomials py(λ) have also been used by C. Lanczos [3]
3 Characteristic values relative to subspaces Let B be an arbitrary (linear)
subspace of S; let 7Γ be the operator on S which carries any vector into its pro-
jection on B. We define a linear operator 4 (B) on B to B as follows:
Then A(Q) is a symmetric operator on B, since 4(B) = πAπ. By the characteristic
roots and vectors of A relative to the subspace B, we mean the corresponding
quantities of 4(6). If B is invariant, then these quantities are characteristic for A
itself. We shall use the following easily verified fact: y is a characteristic vector
relative to B with characteristic value λ if and only if y φ 0, y G B, and (Ay, z)
= λ(y, z) for z C B. By a minimum characteristic vector of B we shall mean a
characteristic vector relative to B with least characteristic value. When no con-
fusion can arise we shall omit the qualifying term "relative/*
LEMMA 3. The j characteristic roots relative to the subspace dj(x) are dis-
tinct and are given by the solutions of
p7 ( λ ) = o .
Each characteristic vector (relative to fly) has a non-null projection on x.
To prove the last statement, suppose that γ is a characteristic vector with
characteristic value λ . If (y, x) = 0, then (y, Ax) ~ (Ay, x) = \(y, x) = 0, and
CHARACTERISTIC VECTORS OF A SYMMETRIC MATRIX 237
(y, A2x) = (Ay, Ax) = λ (y, Ax) = 0, , and (y, A^"ιx) = O From the definition
of Gy it follows that y is orthogonal to this space. But y belongs to this space;
hence y = 0, a contradiction.
The distinctness of the roots now follows. For if two independent characteristic
vectors belong to λ then there is a non-null linear combination orthogonal to x
belonging to λ.
To complete the proof we use the basis (3) of Uy . The matrix representation,
call it Ajy of A(&j) relative to this basis has as element in the (k + l)st row and
(Z + l)st column;
(Aξk,ξι), - i)
Using (2) and the second line of (3) we find that
Ai =
Mo
* 1
0
•
•
* 2
0
o* 2
μ-2
• •
•
0
••• tj-i
Thus, the characteristic roots are the roots of the polynomial
qj(λ)= \\Ij-Aj\ ,
where /y is the /-rowed square identity matrix. Let q$0\) = 1. Direct calculation
shows that qι (λ) = pί (λ), and that the ^y(λ) satisfy the same recursion relation
as the pj(\). Hence the two sets of polynomials are identical. This completes
the proof.
LEMMA 4. Let Vj be the minimum characteristic root relative to &y; that is,
Vj = min. root of pj (λ) (j = 1, 2, ••• , r) .
Then
(6) = vr
2 3 8 W. KARUSH
where \χ is the minimum characteristic root of the invariant subspace uΓ . Further,
each root cr of each polynomial pj(\) satisfies
( 7 ) Kin < σ < λin < σ < λ m a x
The last statement follows at once from Lemma 3 and (1) when we notice that
each characteristic root σ is a value of μ(z) = (z, Az)/(z, z); namely, σ is that
value obtained by replacing z by the corresponding characteristic vector.
To prove (6) we apply (1) to the operator A(dj). Using the fact that (Az, z)
= [A (&j)z, z ] for z in Gy, we find that
= min μ(z) , (z in CL ) .J
From U. d U.-+! we infer that the roots are non-increasing. Suppose that v^
= v\t\\ . Denote the common value by v. From the recursion formula for the poly-
nomials it follows that
contrary to the definition p 0 (λ) = 1.
LEMMA 5. The minimum characteristic vector relative to Uy is given by
where
More generally, the characteristic vector belonging to an arbitrary root σ is
obtained by replacing vy by σ on the right in (8). To prove this, let z denote the
vector obtained by this substitution. It is sufficient to show that η = Az — σz
is orthogonal to Cίy; to this end we use the basis in (3). Using the definition of
z and the relations (2) and (3), we find that
-(σ- μo)]\x\2=O,
CHARACTERISTIC VECTORS OF A SYMMETRIC MATRIX 2 3 9
τί-i Tl
for
obtain
= — V [PI + I M - ίPiWCσ -Mz) - P H W * ! 2 ! ] =0
Z = 1,2, , j — %. For I — j — 1, the term in p/+t does not appear, and we
ain
I 2
- pj(σ) = 0 .
This completes the argument.
4 The iteration procedure. We shall henceforth be dealing with a sequence
\x } of vectors; with each vector we associate the quantities described previ-
ously for an arbitrary vector x. To indicate dependence upon xι we shall adjoin
the superscript i to the symbols denoting these quantities.
Consider an initial vector x° φ 0. By definition r° [= r{x0)] is the dimension
of u [= d(x )] , the smallest invariant subspace containing #°. Since U =
dr0 (x°), according to Lemma 3 there are r° distinct characteristic roots
λ0
relative to Q°; and the corresponding characteristic vectors can be normalized
so that
X = Y i + Yo + •••-!- y r o
All vectors considered below will lie in the invariant space G° Henceforth the
symbols λy and yj will denote the characteristic quantities of this subspace.
To specify the iteration procedure at hand we require, besides x°,the selection
of a fixed dimension s > 1. We remark at this point that the significant case is
that for which the dimension of the invariant space &(xι) at every stage exceeds
s; that is,
2 4 0 W. KARUSH
(9) (i = 0,1,2, •••) .
To simplify presentation, unless otherwise stated it will be assumed that this
condition holds. The trivial case in which (9) fails will be treated at the end of
this section.
Consider now the s-dimensional subspace Q^ = Us (x°) Relative to this sub-
space there is, by Lemma 3, a unique minimum characteristic vector x° -f rj° with
(x°, τ}°) = 0; call it xι. Now form &\ = ds (xι) and select x2 as the unique mini-
mum characteristic vector relative to this space of the form xι + η ι , (χι
9 rjι) = 0.
In general we define # ι + ι as the minimum characteristic vector xι + Tjι, (xι, Ύ]1)
= 0, relative to the subspace CL£. Notice that these subspaces form a chain in
which successive subspaces of index i and i + 1 overlap in xι ι .
LEMMA 6. The sequence \x \ is given by
(τ,')2 4 l h i ,) 1 ί s ' '
where vι is the least root of p | (λ). Further,
(ID v* = μ ( * i + i) .
i4Zso {v } is decreasing; in fact
(12) λj < vi = v/ < z^-x < ••• < vί = μ{xl)9
where v\ is the minimum zero of ρι. (λ).
By Lemma 3 the minimum characteristic root relative to &ι
s is Vs\ It follows
by the definition of # l + l that the equality (11) holds. The relations (12) follow
from Lemma 4, condition (9), and definition. The formula (10) is (8) of Lemma 5
interpreted for x = xι and / = s.
LEMMA 7. /rc ierms of the characteristic basis of Q° we have
(13) * i = oίy, + ( 4 y 2 + + α ^ y r 0 ,
(14) ξj = a\ pfiλjyi + 4pj(λ2)y2 + ~ + a*op/(\.o)yro
( ί = 0 , 1 , 2 , •••; j = 0 r l , *•• , r l ) .
CHARACTERISTIC VECTORS OF A SYMMETRIC MATRIX 241
where
(15) 1 +pjj(λfc)
+ +
Furthermore, α^ = 1
(16) 1 = α j < αf < α? < ••• .
Formula (14) follows from (13) by Lemma 2; (15) is a consequence of (13), (14)
and (10) of Lemma 6. To prove (16) we notice that pι. (λ) (j = 1,2, , s — 1)
is not zero, and has the same sign, at Xi and at vι [since by (12) the least root
of the polynomial exceeds these values] . Hence each term in braces in (15) is
positive; this completes the proof.
We conclude the present section with a consideration of the possible failure
of (9). Suppose that for some first value m of i this inequality fails. Then Cί™ is
an invariant subspace, and the minimum characteristic vector xm ι relative to
this subspace is a characteristic vector of A. Thus Gj?1"1"1 is a one-dimensional
invariant subspace containing only multiples of xm ι . It follows that xι = xm ι
for i > m + l But the argument used in establishing (16) shows that xι — Lyχ9ft
L > 0, for i > m + 1. The theorems to be proved in the next two sections now hold
trivially. We are thereby justified in the assumption of (9).
5 Convergence in direction. We shall first prove that the sequence \xι\
converges in direction; in §6 we shall establish the more troublesome property
of convergence in length.
THEOREM 1. Starting with an initial vector x° ^ 0, and a fixed dimension
s > 1, construct the sequence \x } described above. Then
l i m\yi
Proof. From (12), the sequence \vι\ is a strictly decreasing sequence bounded
from below by λx . Hence there is a number v such that
242 W. KARUSH
lim V1 = V >
By (12) the smaller root v\ of the polynomial pι
2 (λ ) is not less than vι. Hence
pi{vi) = (vi - μ * ) (Vi - μi) - (t\f>0,
(t})a < (^"'-v'KM-v*),
since μι
0 = μ ( x ι ) = V1""1 [see (2) and (11)] . By (1) there is a constant M, inde-
pendent of i, such that
(17) (ίi) 2 < M(vi"1-vi) .
In particular,
ί ι —> 0 as i —> oo.
Recalling (13), put
Thus
Γ, - Σ 6JτΛ
From (14) and the definition of t\, we have
(*ϊ) 2 = ΓTTi = ( M ) a [ « ( λ i ) ] a + ••• + (bjo)2 [ p i ( λ r o ) l 2 .
Since the sum of squares on the right tends to 0, each term must do the same. But
Pi(λy) = (λy — μ^) - (λj — v1"1) —> (λy — ? ) . From the second equation of
(18), it follows that for some index / we have
v = , | 6 } | — » 1 , 6 } — > 0 for j• £ I .
(The last two conditions follow from the distinctness of the λy.)
We propose to show that I = 1. Suppose I l Then
α»
CHARACTERISTIC VECTORS OF A SYMMETRIC MATRIX 2 4 3
Jnl.JiίL . WL ^ oIJ-,1 I'll Ml
Using (12), we have
\ < λj < vi < ή 0 = 1 , 2 , •••, s - l ) .
It follows that p*. ( λ ) has the same sign at λ = λ j , λ l f v ι . Furthermore, s ince
by Lemma 3 this polynomial has only real roots , we have
\P}(\)\ > |pj(λ,)|.
Thus in formula (15) each term in braces for the coefficients a\ and αj is positive,
and each term for a\ is not smaller than the corresponding term for α j . Hence,
for all i, we have
| α l 1 | \a\\
By assumption, α£ = 1, k = 1,2, , r ° . We now have a contradiction to (19).
Thus I = 1.
Since a\ > 0 by (16), we have b\ > 0. Hence
6 j — > 1 , 6 j — > 0 f o r j φ \ .
The theorem now follows from the first equation of (18).
6. The main theorem. Before proving the principal result, Theorem 2, we
establish two lemmas.
LEMMA 8. Let 13 be an invariant subspace with lowest characteristic value
\χ having multiplicity one. Then for x ^ 0 in B, we have
( \ X <r 1 l£(*) l 2 , , s^ Xμ{x) ~ λi < T-T- * —:—~— whenever μ{x) < Λ.2 .\χ
Proof. (An alternative proof, applicable to normal matrices, is given by H.
Wielandt [4] .) Write x in the form (4) where y , y 9 , yι is a complete set of
orthonormal characteristic vectors in B. We let
* =x - , μ* = μ(x*) ,
244 W. KARUSH
and
ξ=i(x) ~Ax ~ μx , ξ* =ξ(x*) =Ax* - μ*x*
From (#*, yx) = 0, we obtain
(r,yi)=o.
From this and {ξ , x*) = 0, we obtain
From the definition of ζ , we have
£* = Λ* - αxλiyx ~μ*x
= ^ - (/i* ~μ)x + (/x* - λ ^ α i y ! .
Hence
0 = (^ , x) =~(μ* -μ)\x\2 + (/*• - λ i ) β ?
Also
0<(ξ*. ξ*) = {ξ\ ξ)= | ^ | 2 + ( / x * - λ 1 ) ( λ 1 - A t ) α ?
from the definition of ξ. Eliminating a\ from the preceding equation, we obtain
Since x* £ B and x* is orthogonal to yj, we have
μ* > k2 .
Hence, whenever μ < λ 2 , the inequality of Lemma 8 follows from the second
inequality above.
We shall eventually show that the sequence of lengths \xι\ converges. To do
this we shall require a bound on the ratio |p ι . (vι) | / τ * . . This is obtained in the
next lemma.
LEMMA 9. Suppose that for all i wte have s < rι\ Then there exists a constant
K, independent of i and /, such that for i sufficiently large we have
CHARACTERISTIC VECTORS OF A SYMMETRIC MATRIX 2 4 5
|pj(^)l <K(τ/)2 0 = 1 , 2 , •", s-1) .
Proof. By Theorem 1, we have μ(xι) — V1 —* λ1# Hence we may confine
ourselves to is so large that, say,
v ^ - λ , < ( l / 2 ) ( λ 2 - λ x ) .
Consider first / = 1. Apply the inequality of Lemma 8 with x = xι, B = (x°. We
find that
By (11), we have
and
Hence
(20)
as desired.
Let
D
1
λ.2 ~"" μ \xι)
Ip ί i
1λ2 - vι~
2<c — •••• ••
λ 2 — kx
t\)2,
( v -
The inequality (20) may be written R\ < K. We propose to show that for some
constant Kι , independent of i and , we have
(21) R) < K^RJ.J2 0 = 2 , 3 , •", s - 1 ) .
This, together with (20), will establish the lemma.
For the remainder of the proof we omit the superscript i. Writing py(λ) as a
product of linear factors, we obtain from (12) and (7) the result that
(22) \p. (y) \<K2\V-Vj\< K2{Vj ~ λj) .
2 4 6 W. KARUSH
In order to estimate the last difference we make use of the minimum characteristic
vector z relative to the subspace fly = (xi
9 Ax1, , A}~1 xι).
We have
μ(z) = Vj .
By (12) we may apply the inequality of Lemma 8 with x = z and B = fl°. Thus
where
ξ{z) =AZ~VJZ.
The vector ξ{z) is orthogonal to fly and lies in βy+i By (3) the vector is a scalar
multiple of ξj. To determine the scalar we use (8) and (2). We find that
Since (v* =) v < Vj < Vj-ι, the above coefficient of ξj does not exceed Rj-χ
(= i?l._j) in absolute value, Vj-ι being the least root of the polynomial. Also
U l 2 > I* I 2 , by (8). Thus
( 2 4 )
The combination of (22), (23), and (24) yields the desired inequality (21).
We turn to the main theorem. The theorem has an obvious counterpart for the
maximum characteristic vector.
THEOREM 2. Let A be a real symmetric operator on a real vector space of
dimension n. Given an initial vector x ψ 0 and a fixed dimension s (1 < s < n),
construct a sequence of vectors {x } as follows: let x1*1 be the unique minimum
characteristic vector relative to the subspace &s(xι) of the form xι + Ύ)1, with
(x\ T)1) = 0. Then xι converges to the minimum characteristic vector in flU°),
CHARACTERISTIC VECTORS OF A SYMMETRIC MATRIX 247
the smallest invariant subspace containing x°. Furthery the vector xι ι is given
by (10), and the least root of ps(M converges to λ t , provided (9) holds. (In the
event that condition (9) fails, the sequence \xl\ is eventually constant, as re-
marked in the last paragraph of §4.)
Proof, By Theorem 1, it is sufficient to show that the increasing sequence
\xι | 2 converges. It is an easy consequence of (10) that
l*i+T=UT Π
where
* » • ; .;.,
By a well-known theorem on infinite products, to prove the desired convergence it
is sufficient to verify that Σ ^ = o c converges. By Lemma 9, this requirement is
reduced to showing that each of the ser ies Σ^°=o ( τ . ) 2 converges. For j = 1,
this ser ies converges by (17). There is, a constant X t such that \AX\ < Kι \x\
Using this inequality and (2), we obtain
\η+1\<Hence we have
< K2+ ή
It follows that for all i we have
ή < 0 = 2 , 3 , ••', s-1) .
The convergence of the remaining series now follows from the convergence for
; = 1. This completes the proof.
REFERENCES
1. M. R. Hestenes and W. Karush, A method of gradients for the calculation of thecharacteristic roots and vectors of a real symmetric matrix. To appear in J. ResearchNat. Bur. Standards.
2. L. V. Kantorovitch, On an effective method of solving extremal problems for quad'ratic functionals, C.R. (Doklady) Acad. Sci. URSS (N.S.) 48 (1945), 455-460.
248 W. KARUSH
3. C. Lanczos, An iteration method for the solution of the eigenvalue problem oflinear differential and integral operators , J. Research Nat. Bur. Standards 45 (1950),255-282.
4. H.Wielandt, Ein Einschliessungssatz fur charakteristischeWurzeln normaler Matrizen,Arch. Math. 1 (1948-1949), 348-352.
UNIVERSITY OF CHICAGO AND
NATIONAL BUREAU OF STANDARDS, LOS ANGELES
ON THE NUMBER OF INTEGERS IN THE SUM OFTWO SETS OF POSITIVE INTEGERS
HENRY B. MANN
1. Introduction. Let A9 B9 be sets of nonnegative integers. We define
A + β = {a + b]a € Af b € B ' By A°9 BΌ
9 we shall denote the union of
A9 B9 and the number 0, by A(n) the number of positive α's that do not ex-
ceed n. We further put
g.l.b. = α ,n
g.l b.
If 1,2, , k - 1 G 4, A ζf 4, we further put
(4) g.l b. = cXi .π ^ n + 1
The real number α is called the density of A9 OLi the modified density9 and α
the asymptotic density of A. Densities of A9 B9 C9 will be denoted by the
corresponding Greek letters α, β, γ, .
Besicovitch [ l ] introduced (X*, and Erdos [2] OLχ.
The author [3] proved: //C = ° + B for B 3 1 and A0 + β° otherwise,
then for all n ^ C we have
(5) C(n) > α * n + β ( n ) .
It was also shown [3I that in (5), (X* cannot be replaced by (X.
Received November 13, 1950.Pacific /. Math. 1 (1951), 249-253.
249
250 HENRY B. MANN
It is the purpose of the present note to improve (5) to the relation
(6) C(n) >0Lιn +B(n) .
The proof of (6) requires only a modification of the proof of (5), but will be
given in full to make the present note self-sufficient.
The inequality (6) immediately yields
(7) 7 > α ! + /3
if C has infinitely many gaps.
Now (7) is sometimes better and sometimes not as good as ErdόV [2j ine-
quality
(8) y > α + β/2
for the case 0ί > β, B 3 1, C = A0 + J3°. (To establish (8) it is really suf-
ficient to assume that there is at least one b° such that b° 4- 1 C B ) However
(7) holds also for C = A0 + B if B 3 1, and for C - A0 + B° without any restriction
o n β .
2. Proof. We shall now give a proof of (6) for the case C — A0 + B, δ 3 1,
and then shall indicate the changes which have to be made if nothing is assumed
about B but if C = A0 + B° . By α, b, c, * # we shall denote unspecified
integers in A9 B9 C, .
Let τ%ι < n2 < be all the gaps in C. Put nr — n, n — n t = d^ for i < r.
If there is one e £[ B such that
(9) a + e + cίi = Πj ,
form all numbers e + dt for which
(10) α + e + d t = n 5 , ί < r , s < r .
Let Γ be the set of indices occurring in (10). Put B = \e + rfs5 se^.
It is not difficult to prove the following propositions.
PROPOSITION 1. The intersection β f l δ * is empty.
PROPOSITION 2. Γλe integer n is not of the form a + e + ds for any s.
Since (10) also implies
(10') α + e + ds = nt ,
ON THE NUMBER OF INTEGERS IN THE SUM OF TWO SETS OF POSITIVE INTEGERS 2 5 1
it follows that β* contains as many numbers as there are gaps in C which precede
n and which are not gaps in A + B U fi*. Hence we have the following result.
P R O P O S I T I O N 3. If B U B* = Bί9 A + Bt = Ci9 then
( I D Ct(n) -C(n) = B i ( π ) - f i ( π ) .
Thus we have proved the following lemma.
LEMMA. If there is at least one equation of the form a + b + c?, = ΠJ9 then
there exists a B\ 3 B such that C± = A + Bx does not contain n9 and such that
(12) d ( n ) ~C(n) = B i ( π ) - β ( n ) > 0 .
Now let C = A0 + β, B 3 l Clearly, n t > 1. The numbers smaller than n\
are either in B9 or of the form n^ ~" α, or of neither of these two sorts. Also
Tii ft β, since C 3 β. Hence we have
(13) Cta) = m - 1 >Λ(ni - 1 ) +β(m) .
Since δ 3 1, we must have ΛI —1 €j! /4, (Λ! ~ 1) >&. Thus, we obtain
(14) C(ni) >
We proceed by induction and assume (6) proved, when n is the /th gap, / < r.
We distinguish two cases.
Case 1: dΓ-i < n^ . Then
C 3 n i - cfΓ-i = α + 6
We now apply the lemma. Let n be the /th gap in Cχ Then j < r9 and we have, by
induction,
(15) d(n) >0Lιn
and, by the lemma,
U6) C r(n) ~C(n) = B
Subtracting (16) from (15), we obtain (6).
Case 2: c/r—]. > n p Now
n — nΓ-i — l >
252 HENRY B. MANN
Hence we have
A(n - nr-x - 1) > CLχ(n - n r - i )
The numbers between nr-γ and n are either of the form n ~~ α, or in B9 or of
neither of these two sorts. But n ££ B hence,
(17) n - n r - ! - 1 > A(n - nr-x - l) +B(n) - β ( n r - i )
> OL^n - nr-t) + β ( n ) - β ( n r - 1 ) .
By induction we have
(18) C(n r-X) = n Γ -i - (r - l ) > Ot^r-i +J5(n r- 1) .
Adding (17) and (18), we obtain (6).
From the proof it is evident that we may obtain the even stronger inequality
( 6 ' ) C(n) > B(n) min
To establish (6) for C = A0 + B° without the restriction B 3 1, we first
remark that in (13) the term A{nχ — 1) can be replaced by A(τiι). The cases to be
distinguished are dr-ι < n t and dΓ-χ > n\ . The proof of Case 1 is then word by
word the same when we replace B by B° and Bι by B J . In Case 2 we have
n — nr~ι ~ 1 > Πi > k ,
so that /4(τι — Λr-i ~1) > αt(w — flr-i); ^ e remainder of the argument remains
unchanged. For C = i4° + S° , we can obtain the even stronger inequality
(6") C(n) > B{n) mm
which again implies the even stronger result
C(n) > max
(n) -h + minΠi<n
ON THE NUMBER OF INTEGERS IN THE SUM OF TWO SETS OF POSITIVE INTEGERS 253
To establish (7), it is sufficient to show that for any set S we have
S(m) S(n)
m n
if m > n9 n €£ S, S(m) — S(n) — m ~~ n. However, this can easily be verified.
Thus if S has infinitely many gaps, then
- . . . -5 (« ) . S(n)cr = lim m i = lim m i .
m n$S n
It thus appears that in (7) we may replace β by
Bin) -lim inf - ^ > β .
n$C n
If C = A0 + B° , we may of course write
Ύ >max (α! + 3 , OC
REFERENCES
1. A. S. BesicoVitch, On the density of the sum of two sequences of integers, J.LondonMath. Soc. 10 (1935), 246-248.
2. P. Erdos, On the asymptotic density of the sum of two sequences, Ann. of Math. 43(1942), 65-68.
3. H. B. Mann, A proof of the fundamental theorem on the density of sums of sets ofpositive integers, Ann. of Math. 43 (1942), 523-527.
OHIO STATE UNIVERSITY
A THEOREM ON THE REPRESENTATION THEORYOF JORDAN ALGEBRAS
W. H. M I L L S
1. Introduction* Let / be a Jordan algebra over a field Φ of characteristic
neither 2 nor 3. Let a —» Sa be a (general) representation of /. If Ot is an alge-
braic element of /, then S^ is an algebraic element. The object of this paper is to
determine the polynomial identity* satisfied by S α . The polynomial obtained de-
pends only on the minimal polynomial of α and the characteristic of Φ It is the
minimal polynomial of S α if the associative algebra U generated by the Sa is the
universal associative algebra of / and if / is generated by (X.
2. Preliminaries* A (nonassociative) commutative algebra / over a field Φ is
called a Jordan algebra if
(1) (a2b)a = a2(ba)
holds for all α, b £ /. In this paper it will be assumed that the characteristic of
Φ is neither 2 nor 3.
It is well known that the Jordan algebra / is power associative;** that is,the
subalgebra generated by any single element a is associative. An immediate conse-
quence is that if f(x) is a polynomial with no constant term then f(a) is uniquely
defined.
Let Ra be the multiplicative mapping in /, a —> xa = ax9 determined by the
element α. From (1) it can be shown that we have
[fiα«δc~] + [RbRac] + [RcRab] = 0and
RaRiRc + RcRbRa + R(ac)b = RaRbc
for all α, b, c £ /, where [AB] denotes AB — BA. Since the characteristic of
Φ is not 3, either of these relations and the commutative law imply (1). Let
Received November 20, 1950.
•This problem was proposed by N. Jacobson.
**See, for example, Albert [ l ] .Pacific h Math. 1 (1951), 255-264.
255
256 W. H. MILLS
a —* S α be a linear mapping of / into an associative algebra U such that for all
α, b9 c C / we have
(2) [SaSbe] + [SbSvc] + [ScSab] = 0
and
(3) SaSbSc + ScSbSa + θ ( α c ) 6 = θ α 56 c + SbSac + ScSab .
Such a mapping is called a representation.
It has been shown* that there exists a representation a —> Sa of / into an
associative algebra V such that (a) U is generated by the elements Sa and (b) ifa > Ta is an arbitrary representation of / then Sa —> Ta defines a homo-
morphism of {/. In this case the algebra V is called the universal associative alge-
bra of /.
We shall now suppose that a —> Sa is an arbitrary representation of /, and (X
a fixed element of/. Let s(r) = Sar,A = s( l) , B = s(2). If we put a = b = c = α
in (2), we get ^ δ = BA. If we put α = b = α , c = αΓ~ 2 , r > 3, then (3) becomes
(4) s(r) =2As(r - l ) + s ( r - 2) B - Λ 2 s ( r - 2) - s ( r - 2 ) A2 .
We now see that.A and 5 generate a commutative subalgebra ί/α containing 5(r) for
all r . By the commutativity of ί/α, (4) becomes
(5) s ( r ) = 2 Λ s ( r - l ) + ( β ~ 2 Λ 2 ) s ( r - 2 ) .
We now adjoin to the commutative associative algebra £/α an element C commuting
with the elements of ί/α such that C 2 = B - 4 2 . We have the following result.
LEMMA 1. For all positive integers r, we have
s(r) = (l/2)(A +C) r + (1/2)(A - C y .
Proof. If r = 1, then
(1/2)(A+C)r
*For a general discussion of the theory of representations of a Jordan algebra and aproof of the existence of the universal associative algebra, see Jacobson [2]
A THEOREM ON THE REPRESENTATION THEORY OF JORDAN ALGEBRAS 2 5 7
If r = 2, then
(l/2)(A+C)r + (1/2)(>1-C) r = 4 2 + C 2 = s ( 2 ) .
Now suppose that r > 3 and that Lemma 1 holds for r — 1 and r — 2. By direct
substitution it follows that A + C and 4 — C are roots of
x2 = 2A* + β - 2A2 ,
and therefore of
xr = 2Axr-1+ (B - 2A2) xr'-2 .r =
Hence,
U + C)r = 2A(A + C)r"ι+ {B - 2A2)(A + C) r"2
and
(A -• C) r = 2A (A - C ) Γ - χ + (β - 2A2 ) (A - C)Γ~ 2 .
Adding and dividing by 2, we have the desired result:
(1/2)(4 + C)r + (1/2)(A-C)r =2As(r - l) + (B ~2A2) s(r - 2) = «(r) .
An immediate consequence of Lemma 1 is that if g(x) is an arbitrary polynomial
with no constant term, then
(6) S g ( α ) = (1/2) g(A +C) + (1/2) g(A - C) .
Now suppose further that GC is an algebraic element of / and that f(x) is a
polynomial with no constant term, such that /(a) = 0. Then by (6) we have
(7) 0 = 2 S / ( α ) = /(A + C) + /(A - C) ,
0 = 2Sα/(α) = (A + C) f(A + C) + (A - C) f(A - C) .
The next step is to eliminate C from the system (7). To do this we need some
additional tools.
3 Theory of elimination. Let Ω be the splitting field of f{x) over the field $ .
Let P = Φ[χ], Q = P[y], P' = Ω[x], <?' = P'[y] be polynomial rings in
one and two variables over Φ and Ω, respectively. Then P and P' are principal
ideal rings. If qx and g2
a r e elements of Q, let (qi9 q2) be the ideal of Q generated
by qί and q2 , and let { x, r2} be a generator of the P-ideal (ql9 q2) Π P. Simi-
larly, if ^ and q2 are elements of Q1 , let ((^i, ^2)) ^ e ^ e ideal of Q' generated
2 5 8 W. H. MILLS
by qι and q2 . Furthermore, let U<7i> <72> > denote a generator of the P ' - ideal
((<7i* #2)) Π P ' . We note that {qi9 q2] and {{qi9 q2}] are determined up to unit
factors. The unit factors are nonzero elements of Φ and Ω respectively.
We shal l establ ish the following lemma.
LEMMA 2. If qι and q2 are elements of Q, then {qχ9 q2] = liqi, #25 5 UP t o
a unit factor.
Proof Let ωi9 ω2 , # , ωm be a basis of Ω over Φ. Then P ' = Σω^ P and
(?' — Σ ω j ^ . Therefore
and
( ( q ! , q 2 ) ) Π P ' =Σωi((ql9q2) Π P) = ( ( g i , g 2 ) Π P) P # = { q i , g 2 } P ' .
It follows that foi, g2} = {{g l f ςτ2Π
Let r and s be distinct elements of P' , and let m and n be positive integers.
We shall determine {{{y - r ) m , (y - s)n]}.
LEMMA 3. Let S(m9n) be that positive integer satisfying
S(m, n) < m + n — 1 ,
and
\n - 1 = 0 i f S(m,n) < N < m + n - 2 ,
where I ..I is the binomial coefficient considered as an integer in Φ. Then we have
U ( y - r ) . (y - . ) " ! ! = ( - r ) 5 ( » ' " > .
Proof. We note that S(m9n) depends only on m, n9 and the characteristic p of
Φ . If p = 0, or if p > m + rc ~ 1, then S(m9n) = m + rc — 1. In any case,
(8) m + n - 1 > S (m, n ) > n .
A THEOREM ON THE REPRESENTATION THEORY OF JORDAN ALGEBRAS 2 5 9
Replacing y by y + r, we may assume that r — 0, s f1 0. Formally, modulo ym
9
we have
μ=0
m-l
1 μ=0
n + μ - 1n - 1
= Σv=n
- 1n - 1
Therefore there exists a q £ Q' such that
(9) qy" + (y-s)"(-l) n £ ss(κ
It follows that
- s ) B Π | s s ( " ' B )
Put
. (y - s ) B Π | s
\ h a , (y ~ s Y \ \ = G ,
Then G and H are elements of P' . Furthermore, there exist qt and g2 \n Q' such
that the y-degree of q2 is less than m and such that qιym + <72(y ~~ 5 ) Γ l ~ C.
Hence
(10) g i ^ y α + g 2 t f (y - « ) n = GH = s
s ( * π ) .
Subtracting (9) from (10) and comparing terms not divisible by y m , we obtain
S(m,n)
(11) q2H=(-l)n
Comparing coefficients of y* 5^' 7 1 ' """ in (11), we get
HS(m,n) - I 1
n - 1 J
2 6 0 W. H. MILLS
which is a nonzero element of Φ. Therefore H is a unit element, and this es-
tablishes Lemma 3.
In the following we shall use l.c.m. (aΪ9 a2, * * * , an) for the least common
multiple of ax, α 2 , * * ' , an.
LEMMA 4. If ((qί9 q2)) Ξ> P' 9 then
Proof. P u t p t = {{qϊ9q3]} , p 2 = U ^ ^ Π , and p 3 = l.c.m. ( p ^ p 2) .
We note that((qlf q3)) DP' 2 (iqιq2* £3)) Π P ' , and therefore P l | {^1^2 > Π
Similarly, p 2 I U^i^2> ^3 I L and hence p 3 | U^i^2> ^ 3 ^ Now there exist D, £,
F, G, H, ΓinQ' such that
Therefore
Dqxq2 +
Hence there exist Kf L, M9 N in Q1 such that
Kqtq2 + Lq3 = p 3 g 2 and
Hence
(fflf + I/f)gig 2 + (fflV + I L ) g 3 = p 3 .
Therefore {{qιq29 qzlW Ps 9 a n ^ t n e proof of Lemma 4 is complete.
We shall now determine ID, El , where
0 = / ( * + y) + / ( * ~ y ) ,
E = (x + y) f{χ + y) + (x ~ y) f{χ ~ y) .
By Lemma 2, we have \D,E\ = U ^ . ^ H - Since
£ - (x - y)D = 2y/(χ + y) ,
we have
+ y ) \ } .
A THEOREM ON THE REPRESENTATION THEORY OF JORDAN ALGEBRAS 2 6 1
Put
lif{χ+y),f(χ-y)tt =Δ.
Let n be the degree of /(#). Choose F(γ) and G(γ) in Q1 , with y-degree less than
n, such that
F(y) f(* + y) + G(y) f(χ - y) = Δ.
Then F(y) and G(y) are completely determined. Now
F(-y) fix ~ y) + G(-y) f(x + y) = Δ.
Therefore we have F( —y) = G(y), from which it follows that F(0) = G(0), or
( F ( y ) - G ( y ) ) / ( * + y) + Giy) D = Δ .
Therefore U θ , y / U + y)}} | Δ . It is clear that Δ | {{D,yf(x + y)\}. Thus we
have
{ D , E \ = { { D . y f i x + y ) ] ] = Δ .
We must now determine
Let/( ie) = Π(Λ; - 0Li)ni, where the 0ίt are distinct elements ofΩ . Then
/(* + y) = Π(* + y - cci)"' , fix - y) = Uix - y - α; )"; .
If qι and q2 are two relatively prime factors of f{x + y), or of f(x "~y), then
((^rt, ^r2)) — P' Therefore we can apply Lemmas 3 and 4 to obtain
(12) \D,E] = Uf(χ+y), f(χ-y)}} = l.cm. (2χ - a* - aj)s^'nJ).
4. The equation for Sα . We shall establish the following result.
THEOREM. Let α be an algebraic element of J satisfying the equation
/(cc) = 0, where f(x) is a polynomial with no constant term. Let
fix) = Π ( * - C ί i ) Π i .
2 6 2 W. H. MILLS
where the 0L( are distinct elements of the splitting field Ω of f(x). Put
φ(x) = l .c.m.(* - (l/2)<Xi- (l/2)αy) 5 ( l | i "J> .
Then φiSa) = 0. Furthermore, if the algebra U generated by the Sa, a £ /, is
the universal associative algebra of J9 if fix) is the minimal polynomial of CL9and
if J is generated by Ct, then φix) is the minimal polynomial satisfied by S<χ.
Proof. As before, we let P = *>ix] 9 Q ~ P[y] be polynomial rings over $ in
one and two variables respectively, and put
D = / ( * + y) + f(χ ~ y)
and
E = (x + y) f{x + y) + (x - y) f{x - y) .
From (7) and (12) it follows that \jj{Sa) = 0. We must now show that φ(x) is the
minimal polynomial of Sα under the three given conditions. If we let (/(%)) be the
principal ideal of P generated by fix), then / is isomorphic to the quotient ring
P/ifix)) under the natural mapping g(θί) —» gix) + ifix))> Let V be the quotient
ring Q/iD,E)* We now consider the linear mapping
(13) g ( x ) _ * Γ g W = (l/2)g(* + y) + ( l/2)g(* -y) + (D,E)
of P into V. By the commutativity of V we have, for all g9 h9 j C P9
(14) [TgThj] + [ThTgj ] + [TjTgh] = 0 ,
since each of the three terms vanishes. Furthermore, by direct substitution we have
(15) 2TgThTj + Tghj = TgTh] + ThTgJ + TjTgh .
We now determine the kernel K of the mapping (13). By definition, gix) G ί if
and only if gix + y) + gix — y) C (D9E). Now
yf(x + y) = (1/2) £ - (1/2) (* - y) D C (D,E)
and
y/(* - y ) = (1/2)(x + y)D - (1/2) £ € (D,E) .
Let gr(%) be an arbitrary element of P. Then, for suitable hix9y) C (?> w e have
A THEOREM ON THE REPRESENTATION THEORY OF JORDAN ALGEBRAS 263
+y) /(* + y) +g(*-y) /(* -y) = ?(*)£ +M*.y) y/(χ +y)
Therefore q(x)fix) C K for all q(x), and thus K 5 (/(*))• Suppose gθt) £ K9
gix) €£ (/(#)). We may suppose that the degree of gix) is less than n, the degree
of fix). Then gix -f y) + g(% — y) = hj) -\- h2E for suitable Ax and A2 in (?.
Since the degree of D is n and that of £ is n + 1, it follows that hγ — h2 — 0.
Therefore g(% + y) + gix ~~ y) is identically 0. This implies that gix) is identi-
cally zero, a contradiction; hence we have K = ifix)). It follows that
g(α)—> Γg(;t) = (l/2)g(x +y) + (l/2)g(x -y ) + (A*)
defines a single-valued linear mapping of / into V\ Furthermore, (14) and (15)
imply that this mapping is a representation, and from (12) it follows that Tx, the
image of OC, has \pix) — {D9 E] as its minimal polynomial. Now since U is the
universal associative algebra of /, the mapping Sg(oς) —> g(x) defines a homo-
morphism * of U into V. It follows that φix) is the minimal polynomial of S α . This
completes the proof.
We conclude by mentioning two simple consequences of the main theorem.
If /(*) =• xn , then ψix) = χs(n>n\ Now (8) yields S(n,n) < In - 1, and we
have the following result.
C O R O L L A R Y 1. // α71 = 0, then S&n~ι = 0.
Similarly, we obtain the following result.
C O R O L L A R Y 2. Let fi<x) = 0, where
Then A(Scχ) = 0, where
A(x)= Π (χ-(l/2)βμ-(l/2)βv).
18 In fact it can easily be shown that this mapping is an isomorphism of U onto V,
2 6 4 W. H. MILLS
Proof. Suppose
where the α, are distinct. Now by (8),
S(rii, Πj ) < Πi + Πj: — 1 <
and
A(χ) = Π ( χ - α i ) n ^ + l V 2 Π (* ~ ;i j>i
Therefore \p{x) | Λ(Λ ) , and the second corollary follows.
R E F E R E N C E S
1. A. A. Albert, A structure theory for Jordan algebras, Ann. of Math. 48 (1947),546-567.
2. N. Jacobson, General representation theory of Jordan algebras, Trans. Amer. Math.Soc, scheduled to appear in vol. 70 (1951).
YALE UNIVERSITY
10
AN APPROACH TO SINGULAR HOMOLOGY THEORY
TlBOR R A D O
I N T R O D U C T I O N
0.1. Given a topological space X, we associate with X a complex R — R(X) as
follows. Let £00 denote Hubert space (that is, the space of all sequences r l 5 ,
rn, of real numbers such that the series r\ + + τ\ + converges,
with the usual definition of distance). For p > 0, let t>0, , vp be a sequence
of p + 1 points in £00 , which need not be linearly independent or distinct, and
let |ι>0, , Vp I denote the convex hull of these points. Finally, let T be a
continuous mapping from |t>0> * * > vp I in^° ^ Then the sequence vo, , Vp
jointly with T is a p-cell of the complex R, and will be denoted by(t>0># * #>
Vp, T) . The group Cp of (integral) p-chains in R is defined as the free Abelian
group with these p-cells as free generators. For p < 0, Cp is defined by Cp = 0
(that is, Cp consists then of a zero-element alone). The boundary operator 3p :
Cp —> Cp~-ι is defined by the conventional formula
i=0
for p > 1. For p < 0, Bp is defined as the trivial zero-homomorphism. Clearly
B3 = 0, and thus R = R (X) is a complex which is obviously closure-finite in the
sense of [ 4 ] . Accordingly, one can define cycles zp, boundaries bp, and so
forth, for R in the usual manner. The homology groups of R are defined by Hp =
, where Zp , Bp denote the group of p-cycles and p-boundaries respectivelyP/ ^P ' WUCJL"C ^p » upinR.
0.2. The complex /?, which was introduced and studied recently by the writer
[6] , differs from the various singular complexes used in previous literature first
in the use of Hubert space. The general practice is to consider continuous map-
pings T from rectilinear simplexes located in any Euclidean space. Instead, we
Received November 8, 1950.Pacific J. Math. 1 (1951), 265-290.
265
266 TIBOR RADO
use Hubert space in its capacity of infinite-dimensional Euclidean space, a pro-
cedure which may of course be adopted in all the various versions of singular
homology theory. The main departure from previous practice lies however in the
fact that no identifications are made in the chain groups Cp of R : two p-cells
(t>o ># " # >vp > T1 ) R f (t>2 * * * # >vp * T ")R a r e e q u a l i f a n d o n l y i f t h e y are identi-
cal, that is, if VQ = VQ, , Vp — υ'p, T' = T" * Thus the complex/? is of
enormous size as compared with previously used complexes. Let us note that
beyond the lack of identifications, R is further increased by the fact that the
points t>0, , Vp occurring in a p-cell (vo, , vp, T) are not required to be
linearly independent or distinct.
0.3. There arises the question of how the homology groups of R compare with
those arising in previous approaches to singular homology theory. In [6] , the
writer proved that the homology groups of R are isomorphic to those of the so-
called total singular complex 5 = S(X) introduced by Eilenberg [3] . Since this
result will be used in the sequel, we shall now give the precise statement of the
main theorem established in [6] . For each dimension p > 0, let us select a funda-
mental p-simplex, with (linearly independent) vertices d0, ?^p For our own
purposes, it is convenient to choose d0, dί9 d2,* as the points (1,0,0,0, •)>
(0,1,0,0, •)> (0,0,1,0, •), * in £Όo. Given then a sequence v0, , vp
of p -f 1 points in £00 > which need not be linearly independent or distinct, there
exists a unique linear map CC: | dQ, , dp \ —* | v0, , vp \ such that <x(dι)
— V(, i = 0, , p. This linear map is denoted by [v0, , vp] . The total singu-
lar complex S = S(X) of Eilenberg [3] may now be described as follows. For
p > 0, a p-cell of S is an aggregate (do, , dp, T) , where T is a continuous
mapping from \do, , dp | into X. The group Cp of (integral) p-chains of S is
then the free Abelian group with these p-cells as free generators. For p < 0, one
sets Cp = 0. The boundary operator Bp : Cp —* Cp-.χ is defined by
for p > 1. For p < 0, 'dp is the trivial zero-homomorphism. The homology groups of
S will be denoted by Hp We have then obvious homomorphisms
p *-* p f{^p 1 'p <-* p f ^ p
defined as follows for p > 0:
AN APPROACH TO SINGULAR HOMOLOGY THEORY 2 6 7
τp{dor~,dp,T)s= (do,'~,dp,T)R ,
<rp(vo,'-,Vp,T)R= (do,' ',dp,T[vo,- ,vp])s .
For p < 0, τp and σp are defined as the trivial zero-homomorphisms. Unfortunately,
Tp is not a chain-mapping. On the other hand, σp is easily seen to be a chain-
mapping, and hence it induces homomorphisms σ*p : Hp —> Hp. The main result
of [6] is contained in the following statement.
THEOREM. The homomorphism σ*p : Hp —> Hp is an isomorphism onto, for
every dimension p.
Since singular homology theory is sometimes thought of only in relation to
triangulable spaces, it may be appropriate to note that the preceding theorem is
valid for general topological spaces. In particular, the space need not be arc-wise
connected.
0.4 In view of the preceding theorem, the complex R appears as an appropriate
tool in constructing singular homology theory. It is of interest to note that the
various complexes used in previous approaches to singular homology theory may
be derived from the complex R by a combination of the following two types of
reduction.
(i) The chain groups Cp of R are replaced by certain subgroups Γp. For
example, one may select Γp as the group generated by those p-cells (vo, ,
vp, T) for which the points vo, , vp are linearly independent. Another sig-
nificant choice may be based upon the concept of a minimal complex studied by
Eilenberg and Zilber [3]
(ii) One selects in Cp9 for each p, a certain subgroup Gp, and one replaces
Cp by the factor group Cp/Gp. From the computational point of view, this amounts
to an identification of elements of Cp which are contained in the same coset
relative to Gp. For brevity, we shall refer to this type of process as an identi-
fication scheme.
In the present paper, we shall study the effect of the various identification
schemes, occurring in previous theories, upon the homology structure of the com-
plex R, It is easy to see that these identification schemes may be reduced to
three basic types. Our result is that one may apply these basic identification
schemes in any desired combination without changing the homology structure of R
(see Theorem 1 in §4.7). As a matter of fact, we obtain an identification scheme
which appears stronger than those previously used (see Theorem 2 in 4.7 and see
268 TIBOR RADO
§5). This leads to some interesting questions, formulated in §6, which seem to
deserve further study.
0.5. It should be noted that the complex R is semisimplicial in the sense
of L3J > and therefore can be used to construct a complete homology and coho-
mology theory.
0.6. Previous relevant literature, as well as further problems arising in this
line of thought, will be discussed in §6 when convenient terminology will be
available. The writer wishes to express his appreciation of the courtesy extended
by S. Eilenberg and N. Steenrod who made available to him the manuscript of their
yet unpublished book [2] . Both technically and conceptually, the study of that
book proved most valuable in preparing the present paper.
1. I D E N T I F I C A T I O N S IN MAYER C O M P L E X E S
1.1. A Mayer complex M is a collection of Abelian groups Cp, where the
integer p ranges from — °° to + °°, together with homomorphisms
dp Cp > Cp-i ,
such that 3 p - ! Bp — 0. Cycles and boundaries are defined in the usual manner.
The homology groups Hp of M are defined by Hp = Zp/Bp , where Zp , Bp are the
groups of p-cycles and p-boundaries respectively. If M, W are Mayer complexes,
then a set of homomorphisms
fp : Cp >C'p
is termed a chain mapping if ^ή/p = fp-ι ^p > where primes refer to the complex
M1
For clarity, we shall write Cp, 3 p , Hp, and so on, to identify the complex
under consideration. In particular, a p-chain of M (that is, an element of C p) will
be denoted by symbols like cp , dp, and so forth.
1.2. We shall now describe the general process of identification in a Mayer
complex M. Let [Gp] be a collection of Abelian groups such that Gp C Cp and
(1) 3p Gp C Gp-X .
Explicitly: if c* € Gp , then d$c$ C Gp-X. Set Cj? = C$/Gp. Thus, the
elements of C™ are cosets relative to Gp . The general element of C ^ is of the
AN APPROACH TO SINGULAR HOMOLOGY THEORY 2 6 9
form {cp}, where this symbol denotes the coset containing the element cp of
Cp. In view of (1), we can then define homomorphisms
by the formula 3™ [cp] = [dpCp]. Clearly ~d™-x <3™ = 0. Accordingly, the system
of factor groups {C!?}9 jointly with the homomorphisms 3 p , constitutes a Mayer
complex m. We shall say that m is obtained by identification, with respect to the
system {Gp}9 from U. The system \Gp\9 satisfying (l),will be termed an identifier
for M We have then natural homomorphisms
TTp i C p * C p
defined by πp Cp = {cp]. Clearly
Thus Up is a chain mapping, and hence induces homomorphisms
defined as follows. If zp is a cycle in M9 then we let [2^ ]j/ denote the homology
class containing Zp . The symbol [z i$ j m is defined similarly. Then Tί^p is given
by
If rr^p is an isomorphism onto for every p , then we shall say that the identifier
{Gp} is unessential. Thus the process of identification with respect to an unes-
sential identifier does not change the homology structure of the complex.
1.3. We shall state presently a convenient condition for the unessential char-
acter of an identifier {Gp}. Let us observe that the condition (1) in 1.2 means
that the homomorphisms Bp, cut down to the subgroups Gp , may be used to turn
the system \Gp} into a Mayer complex which we call G. The complex m9 defined
in 1.2, appears then as merely the complex M mod G in the sense of the general
relative homology theory of Mayer complexes. From this general theory, the condi-
tion for 77*p to be an isomorphism onto, for all p, is well known: it is necessary
and sufficient that all the homology groups of G be trivial. For convenient appli-
cation, we shall now state this condition explicitly.
The condition (ί/). We shall say that the identifier {Gp} satisfies the condition
270 TIBOR RADO
([/) if the following holds: if zίf is a cycle in M such that zp G Gp then there
exists a (p + l)-chain Cp + X G Gp+t such that 3p+! Cp+1 = zp .
We have then the following criterion.
CRITERION FOR UNESSENTIAL IDENTIFIERS. An identifier \Gp] is unes-
sential if and only if it satisfies condition (ί/).
Since the elements of Gp represent those elements of Cp which are, in a sense,
discarded as we pass from the complex M to the complex m, the criterion may be
also worded as follows: discarded cycles should bound discarded chains. In a
special case, this criterion was used by Tucker [ β ] . As mentioned above, the
general criterion is merely a re-wording of a well-known theorem in the relative
homology theory of Mayer complexes (for a comprehensive presentation, see
Eilenberg and Steenrod [2]). For the convenience of the reader, we shall now
outline a direct proof of the criterion.
1.4. Assume first that the identifier {Gp\ is unessential. Take a cycle
(1) z^CGp.
Then 7Tpzp = \zp] = 0, and hence 77 * p [zp']nf = [τrpzp]m = 0. Since π * is an
isomorphism onto, it follows that zp bounds in M:
(2) 4 = 3? + 1 c*+ 1 .
Application of ττp yields, in view of (l), the equation
0 = 77p Zp = TTp'dp+i C p + ι = ^ + l 7 T + C
Thus Ttp+γ cp + t is a cycle of the complex m. Since 77* is an isomorphism onto, we
have therefore a cycle Zp+X such that 77p+12:p+1 differs from the cycle 77p+1Cp+1
only by a boundary. Thus we can write
Ήp+l Zp+l = πp+l c
Now c^+2 is of the form {cp+ 2 i = 77p+2c^+2 Making this substitution, we obtain
z p + l = πp+i c p + l +<^p + 277p + 2 c p + 2 = ^p+lfcp + l + ^ + 2 C
Hence
(3) ^p+lC^p+l ~~ zp + l + ^p + 2 c p+2) = 0
AN APPROACH TO SINGULAR HOMOLOGY THEORY 2 7 1
Now let us consider the (p -f l)-chain
°ί+l = c + i "" z
By (3) we have dp+ι G G p + 1 , while from (2) we have Zp = B "+i </"+! Thus (1)
is seen to imply that zp bounds a chain contained in Gp+ι. In other words, con-
dition (U) holds.
1.5. Assume now, conversely, that condition (ί/) holds. We have to show
that τr*p is an isomorphism onto for every p.
(i) Suppose we have
(1) 77.p[zp]* = 0
for a certain cycle Zp. The assumption means that TTpZp bounds some chain
c™+t. Since c^+ t is of the form {.Op*^ = 77p+i Cp+i > we have
πpZp = 3g + i77jp+1Cp + 1 =
and hence
πp{z$ -3p+iCp+i) = 0 .
Thus the cycle
(2) Zl= z^-B* + 1 c^ + 1
is contained in Gp. Since condition (ί/) is now assumed, it follows that Zp is of
the form
(3) Z £ = 3 j + 1 d p + i
From (2) and (3) it follows that
z = 3 + i ( c + + ίi +
Thus (1) implies that Zp bounds in M, and hence π*p is an isomorphism into,
(ii) Assign now an element [-z£*]m of H™. Now z™ is of the form
(1) z»=\4\=ττp4.
Since Zp is a cycle, we have
0 = BjJzjS = 3j}7Γpc£ = πp.jBJί cM
p .
272 TΓBOR RADO
Hence
Thus Bp Cp is a cycle contained in Gp-i Since condition (£/) is now assumed, we
(4€GP).
p p
have a chain dp such that
Thus Cp — dp is a cycle:
Now we calculate
rM _ ύί _ #p p — p
= \jτpcMp - πpdp]τpcp - πpdp]Λ .
By (1), Ήpc$ = z™, and by (2), π p ^ = 0. Thus finally
π pίzp]* = Op]. .
Thus 77+p is onto, and the proof of the criterion is complete.
1.6. In marked contrast to the general character of the preceding discussion,
the unessential identifiers actually employed in the sequel are of a very special
and restricted type. There arises the question whether there are general con-
structions yielding unessential identifiers in Mayer complexes. The following
comments may be of interest from this point of view. Let M, L be Mayer complexes
and let
(1) fp: CMp-*Clp
be a chain-mapping such that the induced homomorphisms /*p : Hp —» Hp are
isomorphisms onto. In symbols:
(2) f,p : U"p « HL
p .
Let Np denote the nucleus of the homomorphism (1). Since fp is a chain-mapping,
it is immediate that the system [Np] is an identifier.
In view of the strong assumption (2) one may be tempted to conjecture that
{Np} is unessential. The following simple example shows that this is not the
case, even under extremely special and favorable circumstances. Let M be a finite
simplicial complex described abstractly as follows. The group C^ of (integral)
AN APPROACH TO SINGULAR HOMOLOGY THEORY 2 7 3
2-chains of M is the free Abelian group with a single generator ί. The 1-chain
group C t is the free Abelian group with four generators si9 s2 9 s3, s 4 . The
0-chain group Co is generated by α, b, c, d, e. For p ^ 0, 1, 2, the p-chain group
Cp reduces to a zero-element. The boundary relations are as follows:
Bt = s 1 - f s 2 ~ l ~ $ 3 , Bsj = c — 6, Bs2 = α ~ - c , B s 3 = 6 -~ α, B s 4 = e — cί,
Bα = B6 = Be = Bcf = Be = 0 .
We define first homomorphisms /p : Cp —* C 5 as follows:
/2* = 0, fχS1 = / l S 2 = / l S 3 = 0, fιS4 = S! + 5 2 + S 3 ,
/o α = /o 6 = /o c = α, fod = foe = d .
For p ^ 0, 1, 2, of course /p is the trivial zero-homomorphism. Next we define
homomorphisms Dp : Cp —• ^ p + i a s follows:
ΰ o α = 0, D o f c = ~ " 5 3 , Do^ = s 2 , D o ^ = 0, Z)o e = — s 4 ,
For p ψ 0, 1, of course Dp is the trivial zero-homomorphism. One verifies readily
the following facts.
(i) fp is a chain-mapping.
(ii) BDp7p + Dp-iByp = / p γ£ - γ £ , for every p-chain y$ of Λf. Thus
(iii) Let /Vp be the nucleus of Λ,, and let m be the complex obtained from M
by using the identifier {/Vp},in the sense of 1.2. Then the l-dimensional homology
group Hψ of m is infinite cyclic.
(iv) The l-dimensional homology group //f of M is trivial (consists of zero
alone).
Thus M and m have different homology structures, and hence {Np} is certainly
not unessential. And yet, in view of (i), (ii), the induced homomorphisms f*p :
Hp —» tip are isomorphisms onto. In other words, a very plausible method to
obtain unessential identifiers fails even under very special and favorable con-
ditions.
1.7. In dealing with additively written Abelian groups, we shall use certain
familiar conventions. Thus we shall writ© G — 0 to state that the Abelian group G
274 TIBOR RADO
is trivial (consists of a zero-element alone). If A ι, , An are subgroups of G,
then A i + + An will denote the smallest subgroup containing A t , , A2
2. T H E AUXILIARY C O M P L E X K
2.1. The auxiliary complex K, which played an important role in [6] already,
is merely the "formal complex," in the sense of [2] , of £oo taken as a point set.
The complex K is defined as follows. For p > 0, a p-cell of K is a sequence
( υ o , , Vp) of points of E<χ3 which are not required to be linearly independent
or distinct. Two p-cells (v0 , , vp), (wo, * , wp) are considered as equal if
and only if v^ — w^ i = 0, , p. These p-cells are taken as a base for a free
Abelian group, to be denoted by Cp, the group of (finite) p-chains of K. For p < 0,
one defines Cp = 0. For p > 1, the boundary operator
3p : Cp —>Cp-1
is defined by the formula
Clearly 9 3 = 0 . For p < 0, <3p is of course defined as the trivial zero homo-
morphism.
Let {v0 , , Vp) be a p-cell of Kl Treating the points of £Όo a s vectors in the
usual manner, we describe the barycenter b — b(v0,* * *9Vp) of the points v0,
• , Vp by the formula
v0 + ••• + vp
b =p + l
2.2. The following homomorphisms will be used,
(i) The homomorphism 'dp : Cp —> ^n-i > already defined,
(ii) In terms of any assigned point v of £oo, one defines the cone homo-
morphism
hυ
p: Cp->Cp+1 (p > 0 )
by the formula
, • " , vp,v).
AN APPROACH TO SINGULAR HOMOLOGY THEORY 275
For p < 0, hp is the trivial zero homomorphism.
(iii) The barycentric homomorphism
βp Cp^Cp
is defined as follows. For p < 0, βp is the trivial zero homomorphism. For p = 0,
/30 = 1, the identity. For p > 1, βp is defined recursively by the formula
where b is the barycenter of the points v0 , , v γ.
(iv) The barycentric homotopy operator
Pp Cp > Cp+ι
is defined as follows. For p < 0, Pp(v0 , , vp) = 0. For p > 1, pp is defined
recursively by the formula
PP(v0 , •••, vp) = hhp(βp - 1 - p p - ! 3p)(v 0 , # , vp) ,
where b is the barycenter of the points (v0 , , vp).
(v) For p > 1, 0 < j < p — 1, we define the homomorphism.
by the formula tPfj(v0 , , VJ , ι;; + 1 , , vp) - (v0 , , Vj+ι, VJ , , vp).
The operation ίp>y will be referred to as a transposition. Thus "transposition"
means here a transposition of adjacent elements. According to the definition of
equality for p-cells (see 2.1), we have tpfj(v0, , vp) = (v0 , , vp) if and
only if VJ = Vj+χ.
2.3. The following identities hold among these various homomorphisms
(i) V i hP + Λ £ - i 3 p = l (p > 1 ) .
(ii) -dpβp = ySp-j 3p ,
(iii) ^p + i^p + pp-i ^p = /3p ~ 1 ,
(iv) βptPtj=~βp ( p > l , 0 < j < p - l ) .
2.4. If (f0 , , Vp) is a p-cell of K, then j v 0 , , vp \ denotes the convex
hull of the points v0 , , vp (that is, the smallest convex set containing these
276 TIBOR RADO
points). If cp is a p-chain of K, and A is a convex set in Eoo » then the inclusion
Cp C /I is defined to mean that cp can be written in the form
CP = Σ fej(vo.j , ••*, Vp. j ) ,
where the coefficients kj are of course integers, so that |t>o,/> * * *> vp,j I
/ = 1, , n. One has then the following inclusions:
(i) ^P(vo , ' , t>p) C | v 0 , # # # , vpl ,
(") βp(v0 , , v p ) C l^o , ••*, v p | ,
(iii) Pp(vo .*••• v p ) C | v o > , V p L
As a consequence, an inclusion cp (Z A implies that ΉpCp C A, βpcp C A,
PpCp C A, tpjCp C /I. It is understood that the zero chain cp — 0 is agreed
to satisfy the inclusion cp C A for every convex set A.
2.5. For p > 1, an elementary t-chain in K is defined as a p-chain cp which
can be written in the form (see 2.2 (v))
cp = (vo > ###» vp) + tP,j(vo » # Ί *>p)
LEMMA. Given an elementary t-chain
Cp = (t>0, ", Vp) + t p , ; ( t > 0 , * , Vp) ( P > 1 ) t
ίAe following statements hold:
(i) //p = 1, £λeτι 3pCp ~ 0. //p > 1, ίAe^ 3pCp is α linear combination {with
integral coefficients) of elementary t chains d \ v0 , , Vp |
(ii) / 3 p C p = 0;
(iii) PpCp is a linear combination {with integral coefficients) of elementary
t-chains C | v0 , , vp \ .
Proof. The assertion (ii) is an immediate consequence of 2.4 (iv). The as-
sertions (i) and (iii) are readily verified for p = 1. Hence we can assume that
P > 1 .
Proof of (i) for p > 1. Le*t us note that tpfj{vQ, , vp) is of the form {w0 ,
• , Wp), where V( = «;; for j ^ /, / + 1, and ι?y = wy+i » t>y+i = w y. Now we have
AN APPROACH TO SINGULAR HOMOLOGY THEORY 2 7 7
Pp cp ~ 2* v — J- J l Λ v o > ι v i > » v p y ' V^o > i w i i* * iwp) 1
t=O
For i φ j9 j + 1, the quantity in square brackets is clearly an elementary ί-chain
^ I vo 9 * * * 9 vp I 0 n t n e other hand, the terms corresponding to i = / and
i = / + 1 cancel. Thus (i) follows.
Proof of (iii) for p > 1. Since (iii) is verified directly for p = 1, we proceed
by induction. Assume (iii) to hold for p — 1, where p > 2. Let us write again
ίp,/ = (^o ># * * 9 wp) Clearly, the points v0, , vp and the points w0, , Wp
have the same barycenter ό. Hence we have (see 2.2 (iv))
Pp {v0 , , vp)= hp [βp (v0 , , Vp) - (t;0 , , Vp)
= A
In view of (ii), addition yields
(1) Ppcp = hp( ~cp - Pp-i'dpCp) .
Now, by (i), 3pCp is a linear combination (with integral coefficients) of elementary
ί-chains C (v0 , , vp \. Hence, by the inductive assumption, the same holds
for yOp-i ^pCp9 and hence also for the quantity in parentheses in (1), and finally
for pp cp itself, since b C | v0 , , vp | .
2.6. For p > 1, an elementary d-chain in K is defined as a p-cell (ι;0 , , vp)
such that VJ = vy+i for some /.
LEMMA. // Cp = (vo, , Vp) is an elementary d-chain, then the following
statements hold.
(i) If p — 1, ίAew ^pCp = 0. //p > 1, ίAew 3pCp is a linear combination (with
integral coefficients) of elementary d-chains C | v0 , , Vp \.
(ii) βpcp = Q.
(iii) PpCp is a linear combination (with integral coefficients) of elementary
d-chains C | vQ , , vp | .
The proof is entirely analogous to that in 2.5, except that (ii) requires an
additional remark. We have VJ = Vj+t for some j by assumption. For this same j,
278 TIBOR RADO
we have then the relation
Hence we have also
βp tp9j(vOt 'fVp) = βp
On the other hand, 2.3 (iv) yields
Hence 2βp{v0, , v p) = 0. Since βp(v0, , v p) is an element of the free
Abelian group Cp , it follows that βp (v0 , , vp) ~ 0.
3. T H E C O M P L E X R = R(X)
3.1. In working with the complex R (see 0.1), the following device (introduced
by Eilenberg and Steenrod in [2] in connection with the complex S; see 0.3) is
useful. Let A be a convex subset of £Όo , and let Cp denote the subgroup of Cp
(see 2.1) generated by those p-cells (v0 , , vp) of the complex K which satisfy
the inclusion (v0 , , vp) C A (see 2.4). For p < 0, we define Cp = 0 (see
1.7). Let T : A —> I be a continuous mapping. We can define then homomor-
phisms
Γ # pk v pRp . L*p ' Li p
by the formula
7p(vo, ,vP) = (vo,~',Vp,T)R ( p > 0, (t/0, ,fp) C C^) .
For p < 0, Tp is the trivial zero-homomorphism. For Cp ζL Cp, it will be con-
venient to use the symbol (cp, T) to denote TpCp. Among the simple and obvious
rules of computation for the symbol (cp ,T) , we mention the formula
In terms of the preceding notations, we define now homomorphisms
Hp ' ^ p * v p 9
^.R . r* R v n RHp ^ p f ^ p + 1
AN APPROACH TO SINGULAR HOMOLOGY THEORY 2 7 9
by the formulas
βp(vo,'~,vp,T)R= (βP(vQ,~ ,vp),T)R, ( p > 0 ) ,
Since βp(v0, , vp) C | v0 , , vp | , pp(t>0 , , vp) C \vo, , vp\ hy
2.4, the homomorphisms βp, pp are well defined. For p < 0, /3p and pp are
defined as the trivial zero homomorphisms. In terms of the homomorphisms tp ,
defined in 2.2, we define
by means of the formula
tpj(vo,'",vp,T)R= (tPfj{v0, ~,vp),T)R .
We have then the following identities (see [6]) :
(1) 3«/3« = $-,3";
(2) 3 « + 1 / 0 « + p*^** = βR
p-l,
where 1 denotes the identity transformation in Cp; furthermore (see 0.3)
(3) # t * , , =-/3* ( 0 < J < P ) ;
(4) σ p τ p = l ;
(5) B^σp-^Jrp;
(6) Cp-i^pTpVp = CΓp-i p
(7) σp β« rp σp= σp β«
(8) σ-p+1p*τpσp=σp+1pp
ι;
(9) Tp_ 1 σ p _ 1 3p ί
/ S ρ
ί = B « r p o - p / S « .
3.2. For p > 1, we define an elementary t-chain in R as a chain of the form
(ι»o, " . vp, Tf + (tPfj(v0 ,'• >, vp), T)R (see 2.2). The subgroup of C* gener-
ated by the elementary ί-chains will be denoted by Tp . For p < 0, we define
T* = 0 .
280 TIBOR RADO
LEMMA. If C$ € Γp
Λ, then
(i) ^PcR
pCT^l
(ϋ) / 3 * c * = 0 >
(in) P p C j ? C ^
Proof. Clearly, it is sufficient to consider the case where cp is an elementary
ί-chain:
4= (vo,'",vp,T)R + (tPlj(v0,'",vp),T)R
= ((«o, •• ,vp) + tPtj(v0, ",vp),T)R .
Then we have
tf 4= (Bp[(iΌ. ,Wp) + tp,j(vo, ",vp)],T)κ.
By 2.5 (i), 'dp [(v0, , vp) -f tptj{v0 , , ι θ ] is either zero or else a linear
combination, with integral coefficients, of (p — l)-chains of the form (wo, ,
Wp-ι) + ^-1,^(^0 > * * *> Wp-i)» a^l c l^o ># * #» v p I > a n ( l thus (i) is obvious.
In a similar manner, (ii) and (iii) follow from 2.5 (ii) and 2.5 (iii).
3.3. For p > 1, we define an elementary d-chain in R as a p-cell (v0, * ,
vp, T) such that t>y = Vj+χ for some y, 0 < j < p — 1. The subgroup of Cp gener-
ated by the elementary d-chains is denoted by Dp . For p < 0, we define Dp = 0.
L E M M A , //cp G Dp, then
(ϋ) ^ c/J = 0 f
These statements are immediate consequences of 2.6 (i), 2.6 (ii), 2.6 (iii).
3.4 Given a p-cell (v0, , vp, T)R, take a sequence tυ0, , wp of p + 1
linearly independent points in E& . Then we have a linear mapping Cί: |w;0 , ,
Wp I —» I VQ , , Vp I such that &(wi) — v, , i = 0, , p. Then the p-chain
(1) 4= (vo,~',vp,T)R- {wQ," ,w
AN APPROACH TO SINGULAR HOMOLOGY THEORY 281
will be termed an elementary a-chain. The subgroup of Cp generated by the ele-
mentary a-chains will be denoted by Ap. For p < 0, we define Ap = 0.
LEMMA, C* G Ap if and only ifσpcp = 0 (see 0.3).
Proof. Assume Cp C Ap, Then c ί is a linear combination of chains of the
form (1), and hence it is sufficient to show that crpCp = 0 for the chain (1). Now
we have (see 0.3)
o-C = (do "dTlvo9mΛV])S ~ (rfo rf
Clearly [v0, , vp] = a [w0, , wp], and thus σpCp = 0.
Assume next that CpCp = (
can be written as a (finite) sum
Assume next that CpCp = 0. Then we also have TpσpCp — 0. The chain Cp
(2) 4= Σ njivo.j.—.Vp.j.Tj)*,j
where the coefficients nj are integers. We have then
(3) 0=T p <r p c*= ]Γ nj(dormm,dp,Tj[vo.j,"m,Vp,j'])*
j
Subtracting (3) from (2), we see that Cp appears as a linear combination of ele-
mentary a-chains, and thus cp £ Ap. If p < 0, then the lemma is of course
obvious.
3.5. LEMMA. If C* C A*, then
(i) δ£c£ CΛp-i
(ϋ) fiξc* CA«,
(ϋi) p$c* CA$+X.
These statements are immediate consequences of the identities (6), (7), (8) in
3.1, in connection with the lemma in 3.4. For example, to prove (iii), we note that
by (8) in 3.1, we have
(1) OΓp+i/ pcj} = σp+tpβrpσpc^ 0 ,
since Cp G Ap, and hence CpCp = 0 by 3.4. Also by 3.4, the relation (1) implies
282 TIBOR RADO
that pRc$
3.6. Let us observe that the chain groups Cp, Cp are free Abelian groups by
their very definition (see 0.3) and hence they do not contain elements of finite
order.
4. UNESSENTIAL IDENTIFICATIONS IN R=R(X)
4.1. LEMMA. Let {Gp} be an identifier for R (see 1.2, 0.1) such that the
following conditions hold:
(i) c*£ Gp implies that β* c$ = 0
(ii) cjj £ Gp implies that p$Cp£Gp + x .
Then {Gp} is unessential (see 1.2).
Proof. We shall verify that {Gp} satisfies condition (V) of 1.3. Take a cycle
Zp G Gp. In view of (i) and (ii), the homotopy identity
itf4 = βU - 4a) H+XPUI +
yields the relation
Thus z p is the boundary of the (p + l)-chain Pp Zp £ Gp+χ , and condition (U)
is established. By the criterion in 1.3, it follows that {Gp} is unessential.
4.2. LEMMA. Let {Gp} be an identifier for R, such that the following con-
ditions hold:
(i) Gp D A £ (see 3.4);
(ii) Cp CGp implies that crpβpCp = 0 (see 0.3);
(iii) Cp£Gp implies that ppCp CGp+i
Then {Gp} is unessential.
Proof. Again, we verify that {Gp} satisfies condition (ί/). Let us take a cycle
Zp £ Gp; we have to show that it is the boundary of some chain in Gp+i. We
note that
AN APPROACH TO SINGULAR HOMOLOGY THEORY 283
is a cycle, and that by (ii) we have
CΓpil =σpβ«zR
p = 0 ,
since Zp G Gp. Since σ*p : Hp —> Hp is an isomorphism onto (see 0.3), it
follows that ζp bounds:
(2) ζ j = 3 ? + i 7 p + i .
Applying σp on the left, we get (see 0.3)
0 = σ > ζ * = σ p 3 * + 1 y * + 1 = ^ ^
Thus σp+ιγp+ί is a cycle:
(3) σ+iΎ+i = ^
Since cr+ is an isomorphism onto (see 0.3), there exists a cycle Zp+i such thatzρ+ι a f ld CΓp+i Zp+i differ only in a boundary:
| 1 p + 1 £ 1 - f
- Zp + l ~ ^p + 2 ^ + 2 C + 2 ) = 0 .
Since 3p+2 = crp+1 3p+2 Tp+2 , the relations (3) and (4) yield
(5) <7>+l(7p
On setting
< 6> 4 + 1 = 7 p + l - ^p+l -
we see that the relations (5), (1), (2), (6) yield
(7) cr p + 1 d j + 1 = 0 ,
(8) βR
pz$ = 3 j ϊ + i ^ + i .
From the homotopy identity 4.1 (1) and from ,(8) we infer now that
(9) z =
By (7), (i), and 3.4, we have d*+ι G G p + 1 . Since p^z^ G Gp+i by (iii), it
follows from (9) that zp is the boundary of a chain in Gp+ι, and the proof of the
2 8 4 TBBOR RADO
lemma is complete,
4.3. LEMMA. Let \GΛ be an identifier for R which satisfies the assumptions
of the lemma in 4.1. For each p, let Gp denote the division-hull of Gp. Then \Gp\
is again an identifier (see 1.2) which satisfies the assumptions of Lemma 4.1.
Proof. Take a chain Cp C Gp. Then there exists an integer n φ 0, such that
ncp £ Gp and hence (since {Gp} satisfies the assumptions of Lemma 4.1)
(1) nβ» 4=0,
(2) np«4CGp+1.
C £ G+ Since βcBy the definition of Gp+i, (2) implies that Pp Cp £ Gp+i Since βpcp is an
element of the free Abelian group Cp (see 3.6), (1) implies that βpCp = 0.
4.4. LEMMA. Let \Gp\ be an identifier for R which satisfies the assumptions
of Lemma 4.2. Then {Gp} is again an identifier which satisfies the assumptions
of the same lemma.
The proof is the same as in 4.3, except that one uses now the fact that σpβp cp
is an element of the free Abelian group Cp ^see 3.6).
4.5. LEMMA. Let {G^}, ,{Gίn)} be identifiers for R, satisfying
the assumptions of Lemma 4.1. Then {Gp + + Gp } is again an identifier
which satisfies the assumptions of Lemma 4.1.
The proof is obvious.
4.6. LEMMA. Let Ω' be a collection (perhaps empty) of identifiers for /?,
each of which satisfies the assumptions of Lemma 4.1. Let Ω" be a nonempty
collection of identifiers for /?, each of which satisfies the assumptions of Lemma
4.2. For each p, let Gp denote the smallest subgroup of Cp containing the groups,
with the same subscript p, of the identifiers contained in Ω' and Ω". Then {Gp}
is an identifier satisfying the assumptions of Lemma 4.2.
The proof is obvious.
4.7. The preceding lemmas, combined with the results of §3, yield a number
of unessential identifiers for R. In the following two theorems, the symbols Ap ,
Dp, Tp have the meanings explained in §3.
AN APPROACH TO SINGULAR HOMOLOGY THEORY 2 8 5
T H E O R E M 1. Each one of the systems [A*], {D* } , {T*}9 {A* + D ^ j ,
\Ap + T*}, {D* + Tp], {A* + D* + T*} is an unessential identifier for R
(see 1.2).
THEOREM 2. If ΓJJ denotes the division-hull of the group Γ* = A* + θ£ +7]?,
then {Γp 5 is an unessential identifier for R,
Proof* By 3.5 and 3.4, the system \Ap } is an identifier satisfying the as-
sumptions of Lemma 4.2. Similarly, the systems {Dp }, {Tp} are identifiers
satisfying the assumptions of Lemma 4.1, by 3.2 and 3.3 respectively. By 4.5 it
follows then that {Dp + Tp } is an identifier satisfying the assumptions of Lemma
4.1. Similarly, by 4.6 it follows that U p + Op + T{ }, {A* + θ j } , {A{ + Γj}
are identifiers satisfying the assumption of Lemma 4.2. Finally, [Γjf } is an
identifier satisfying the assumptions of Lemma 4.2, as a consequence of 4.4. The
unessential character of all these identifiers is then a direct consequence of 4.1
and 4.2 respectively.
REMARK. The writer was unable to determine whether or not Γp coincides
5. T H E COMPLEX Γ = r(X)
5.1. Theorem 1 in 4.7 shows that any combination of the basic identification
schemes, used in previous approaches to singular homology theory, may be applied
to the singular complex R without affecting its homology structure. From the point
of view of achieving maximum reduction, the identifier {Γp } is of special interest.
We shall therefore go into some detail concerning this particular identifier. By the
general remarks made in §1, this identifier leads from the singular complex R to
a new and much smaller Mayer complex which we shall denote by r = r(X) Since
{Pp ] is unessential, r has the same homology structure as R. We want to examine
in some detail the computational facilities and conveniences available in the
complex r.
5.2. By the general remarks in §1, the elements of the p-chain group CTp of r
are of the form {cp}9 where this symbol denotes the coβet (relative to Γp) con-
taining the p-chain Cp of R. Let us adopt, in dealing with the complex r, the usual
practice of writing Cp instead of {cp}, with the understanding that Cp is now
considered as a representative of the element {cp} of CTp. For clarity, we shall
use the congruence symbol = in writing equations, to remind ourselves of the
286 TABOR RADO
fact that we are dealing actually with congruences mod Γp . We shall presently
note some of the computational rules for the complex r.
5.3. Let (v£, , v'p\ T')R, (v'όr ••, v£, T")B be two p-cells of R related
as follows. There exists a system of linearly independent points w0 , , Wp in
£ 0 0 a n d t w o l i n e a r m a p s α ' : \wo, ,wp\ — > \ V Q 9 9 Vp \9 O ί " : \w0,
• , Wp I —» I VQ , , Vp I, such that the following relations hold:
(i) α'(»i) = v'i , α"(»i) = «"i (i = 0 , , P ) ,
(ii) r α' = Γ"α" .
Then {vi , - , Vp , T')" = ( < , , Vp , T" ) R . Indeed, by the definition of A*
and Γp (see 3.4, 4.7), we have
(v'o,' ',Vp,τ')R ~ (y>o," ,vp,T'a') f
and hence
Similarly
Since Γ' α' = Γ ' ' ^ , the assertion follows.
5.4. Given a sequence v0 , , Vp of p + 1 points in £00 (which need not be
linearly independent or distinct), by a transposition we shall mean (as in §2) the
operation of exchanging two adjacent elements of the sequence v0 , , vp. Let
then (VQ, , Vp, T' ) R , (I/Q , * #» fp\ 71")* be two p-cells related as follows:
(i) \vΌ,'~,v'p\ = K •••,!/; I ,and T' = Γ "
(ii) there exists a sequence of n > 0 transpositions leading from {v'Q , ,Vp)
to (t/s , . . . , * ; ) .
T h e n K , , t ; ^ , Γ ' ) Λ Ξ ( ^ , . - - , V £ , T " ) R if n i s e v e n , and ( ^ , ,
Vp, T')R = -(V'Q, ,v'p, T")R iί n is odd. Indeed, the assertion is obvious
if 7i = 0. If n — 1, the assertion follows immediately from the fact that Tp C Γp
(see 3.2, 4.7). Repeated application of this remark yields the desired result for a
general n.
AN APPROACH TO SINGULAR HOMOLOGY THEORY 2 8 7
5.5. Let (vo, , Vp, T) be a p-cell such that the points vQ, , Vp are
not all distinct. Then (ι;0 , , Vp, T) = 0. Indeed, by a certain number n of
transpositions we can obtain a p-cell (u/0, >Wp» T) in which two adjacent
points WJ , Wj+X coincide. Then (see 3.3, 4.7)
(wo,--,wp, T)R C D* C f* ,
and hence
(^0, , ^ , Γ ) Λ Ξ 0 .
On the other hand, by 5.4,
(«'o, ,«'p,Γ)/1 s ±{vo, ",vp,T)Λ,
and the assertion follows.
5*6. Let (vQ , , Vp , T) be a p-cell of /?. Let w0 , , u;^, where q > p,
be a system of linearly independent points in E& > and let α : | M ; 0 , ,w;^|
—> I v0 , , Vp I be a linear map such that the points 0((M;0), , (Xdi; )
coincide with the points t>0 , , vp in any order and with any number of repe-
titions. Then
(»ormm,Vq,TOL)B Ξ 0 .
Indeed, by 5.3 we have the relation
(•o. ' . ϊ .Γα) ' 5 (a(wQ), ~,u.(wq),T)R.
On the other hand, since q > p, the points <x(wo)9 , &(wq) are not all distinct.
Hence, by 5.5, we have
and the assertion follows.
5.7. Let (v0 , , Vp, 71) be a p-cell of /?, such that the points v0 , , vp
are linearly independent. Suppose this p-cell possesses the following type of
symmetry. There exists a linear map (X : | v0 , , vp \ —> | v0 , , vp \, such
that (i) the points Ot(t;0), , (X(vp) form an odd permutation of the points v0,
• •> Vp (taken in the indicated order) and (ii) Γ(X = Γ. Then (v0, , vp, T)
= 0. Indeed by 5.4 and 5.3 we have
288 TIBOR RADO
(vo,' '',vP,T)R =-(α(υ o ),
Since T = TOL, it follows that 2(v0, , vp, T)R = 0, or equivalently
2(^o, %^,Γ)/iG ΓR.
Now since Γjf is the division-hull of Γp (see 4.7), the last relation implies the
existence of an integer k ψ 0 such that 2k (v0, , vp, Γ) G Γp , and hence
(by the definition of the division-hull) (v0 , , vp, Γ)Λ G Γp . Thus (v0 , ,
Vp, T)R = 0.
5.8. The argument just used yields obviously the general result: if ncp = 0,
where n is an integer ^ 0, then cp = 0. In other words, the p-chain group Cp of
the complex r has no elements of finite order. Of course, this is a priori obvious
from the remark that a division-hull is closed under division. It may be of interest
to determine whether or not Cp is in fact a free Abelian group. The writer was
unable to answer this question.
5.9. The homomorphisms Bp, βp9 pp , τpσp apply to congruences. In detail:
OΌ Cp = O« Cp, Pp Cp — Op Cp f pp Cp = pp Cp , ^Ό&Ό Cp = 'p&p Cp .
The first one of these asserted congruences is of course merely a restatement of
the fact that \Γp } is an identifier. The last one may be verified as follows. In
view of the identity 3.1 (4) we have
4 - 4 ) = σP 4 ~ σP CP =
and hence, by 3.4,
τpσp cp-cR
p CA*.
Since Ap C Γp , it follows that
(1) = Cp
Similarly, rpσpcR = cR. Since cR = cR, it follows that rpσpc
R = τpσpcp. Now
let us recall that μ^} satisfies the assumptions of Lemma 4.2, as we observed
in the course of the proof in 4.7. Accordingly, the assumption cp = cp , which is
equivalent to Cp — δ R £ Γjf , implies that
AN APPROACH TO SINGULAR HOMOLOGY THEORY 2 8 9
(2) crpβ«(c«-c«)=0,
(3) tyThe relation (3) is equivalent to PpCp = ppCp. On the other hand, (2) implies,
by 3.4, that βR
p(c$ - δ*) £A* C f/, and hence that β*c* = β*c$.
5.10. In terms of familiar terminology, the preceding results may be summarized
as follows. In the complex r, affine-equivalent p-cells of R become equal to each
other (see 5.3). The permutation rule (or the orientation convention) holds in r
(see 5.4). Degenerate p-cells of R may be discarded in r (see 5.5, 5.6), as well as
affine-symmetric p-cells (see 5.7). The operators 'dp, βp, pp continue to apply
in r (see 5.9). Furthermore, the operation τpσp is also applicable in r (see 5.9).
The effect of this operation is to replace a general p-cell (v0, , vp, T) by a
p-cell of the form (do,0 *9dp, T*) (see 0.3). Accordingly, one can avoid en-
tirely the use of p-cells (v0 , , Vp , T) where the points v0 , , vp are not
linearly independent (it is not obvious, however, that this practice, if followed
consistently, contributes to clarity and simplicity of calculations). Finally, let
us note that the complex r offers the advantage that its chain-groups do not have
elements of finite order (see 5.8). In the light of comments made in previous liter-
ature, this may represent a desirable feature.
5.11. In the course of a correspondence on these subjects, Professor S.
MacLane communicated to the writer a simple and ingenious proof of the fact
that the chain-groups of the complex r are indeed free Abelian groups (cf. 5.8).
6. CONCLUSION
6.1. One may raise the question whether the singular complex R admits of
further reductions, in terms of identifications, without affecting its homology
structure. In particular, one may ask whether there exists a maximal identification
scheme, in some natural and appropriate sense. A plausible approach may be
obtained by setting up the principle that only those identifications are admitted for
which the computational rules set forth in 5.3—5.9 hold. The problem consists then
of determining whether among all unessential identifiers {Gp}9 conforming to this
principle, there exists one, say {Gp}9 such that Gp C Gp for all identifiers {Gp}
satisfying the requirements just stated. The writer was unable to settle various
interesting questions upon which the answer to this problem seems to depend.
6.2. From a heuristic point of view, one may conjecture that, in view of the
intensive study and manifold applications of singular homology theory, it is un-
likely that any relevant identification scheme escaped the attention of the many
290 TffiOR RADO
workers in this field. For example, one may assume, as a heuristic working hy-
pothesis, that by applying simultaneously all the identification schemes used in
the papers listed in the References of the present paper one obtains a maximal
identification scheme in the sense of 6.1. The writer was unable to find a proof
for the theorem suggested by these remarks.
6.3. As regards previous literature concerned with the unessential character
of identification schemes, precise comparisons would lead to excessive detail,
particularly because our complex R has not been considered explicitly in the
literature, as far as the writer is aware. The following comments are meant to
indicate the origin of certain questions rather than the exact formulation of defi-
nitions occurring in other theories. The initial motivation for the present study,
as well as for the previous paper [6] of the writer, came from the important paper
of Eilenberg [ l] In that paper, Eilenberg shows, in effect, that (in our termi-
nology) the identifier {Tp } is unessential (see 3.2). In his previous paper [6] ,
the writer showed then that the identifier \Ap } is also unessential. However, the
unessential character of certain identifications has been recognized by various
authors. Thus Seifert-Threlfall [7] and Lefschetz [5] contain remarks suggesting
that the "affine symmetric " p-cells may be discarded without affecting the homol-
ogy structure. Tucker [δ] showed, in effect, that the system {Dp} is unessential,
at least in relation to the identifier {Tp ]. In a sense, our complex R appears thus
as the singular complex in unreduced form, alternative theories being derivable by
various types of reduction. The problems we stated in 6.1 and 6.2 amount merely
to the question whether there is some end to this process of reduction without
changing the homology structure.
REFERENCES
1. S. Eilenberg, Singular homology theory, Ann. of Math. 45 (1944), 407-447.
2. S. Eilenberg and N. E. Steenrod, Foundations of Algebraic Topology (Unpublished).
3. S. Eilenberg and J. A. Zilber, Semisimplicial complexes and singular homology,Ann. of Math. 51 (1950), 499-513.
4. S. Lefschetz, Algebraic Topology, Amer. Math. Soc. Colloquium Publications, vol.27; American Mathematical Society, New York, 1942.
5. , On singular chains and cycles, Bull. Amer. Math. Soc. 39 (1933), 124-129.
6. T. Rado, On identifications in singular homology theory, To appear in Rivista diMatematica della Universita di Parma.
7. H. Seifert and W. Threlfall, Lehrbuch der Topologie, B. G. Teubner, Leipzig, 1934.
8. A. W. Tucker, Degenerate cycles bound, Rec. Math. (Mat. Sbornik) 3 (1938), 287-289.
OHIO STATE UNIVERSITY
ON SOME TRIGONOMETRIC TRANSFORMS
O T T O SZASZ
l Introduction. To a given series Σ ^ = ι un we consider the transform
A sin vtnAn = 2* uv > where tn Φ 0 as n —* °°
It was shown in a previous paper [5, Section 4, Theorem 3] that the transform
(1.1) is regular if and only if
(1.2) ntn = 0 ( 1 ) , as n• o o
We shall now consider the transform (1.1) in relation to Cesaro means. In a forth-
coming paper Cornelius Lanczos has found independently that the transform (1.1)
is very useful in summing Fourier series and derived series, and gave some very
interesting examples; he takes tn — ττ/n. Of our results we quote here the follow-
ing theorem:
THEOREM 1. In order that the transform (1.1) includes {CfD summability, it
is necessary and sufficient that
(1.3) ntn=pπ + (Xn, n<Xn = θ(l), p a positive integer.
We also discuss other triangular transforms which may be generated by "trun-
cation" of well-known summation processes, such as Riemann summability. The
transform An and the transform Dn (Section 5) are special cases of the general
transform
n
Ύn = Zs
Received March 8, 1950. Presented to the American Mathematical Society December 30,1948. The preparation of this paper was sponsored (in part) by the Office of Naval Research.
Pacific J. Math. 1 (1951), 291-304.
291
292 OTTO szXsz
where φ(P) is a function of the π-dimensional point P(xι, #2> * ' *> xn)>
Pn —> 0. This transform and many special cases of it were discussed by
W. Rogosinski [4] in particular, the special case an = 0 of our Theorem 4 is
included in his result on page 96. The general approach is essentially the same
as in the present paper.
2 Proof of Theorem l If we write
" " , sinvtn sin (y + l) tn
sin vtn 2 sin {v + l) tn sin (v -f 2) t n _
(v + 1 ) ίn (v + 2 ) t n
I
then
n t
= " y s ' Δ 2 + s ' Δ + ( ' - s ' ) —
or
/0 ,x . _ n ^ 2 , Λ2 , , I sin (n - 1) tn 2 sin ntn
ntr,
sin
nntn
Now (C. 1) summability of 2 n = 1 un to 5 means that
(2.2) n - i s ^ —> s , as n —• 00 .
If sn = 1, then i4Λ = sin ίΛ/ίπ —> l
In order that (2.2) imply An —> 5, it is necessary and sufficient [in view of
(2.1)] that
ON SOME TRIGONOMETRIC TRANSFORMS 2 9 3
sin ntn , x sin (n — l) tn(2.3)
n-2
(2.4) Σ H Δ H = 0 ( 1 ) , as n
v=l
The first condition of (2.3) [in view of (1.2)] is equivalent to
sin ntn = 0(tn) = 0(l/n) ;
hence
ntn = pπ + an , ndn = θ(l) .
The second condition of (2.3) now reduces to
cos ntn sin ίn = θ ( ί n ) ,
or
cos α n sin ίn =θ(n"1) ,
which is satisfied. Finally
= / cos vx dx = K / e ι c/x
hence
(2.5) t n Δ ^ = R j Γ t n tfeivx dx=Hfo
ta eivx(l-eix)2 dx ,
and
(2.6) t B I Δ i I < JΓ t n |1 - β " | 2 dx = 4 jftn (sin x/2)2 dx
It follows that
n-2
294 OTTO szXsz
This proves Theorem 1.
We can show by an example that the transform An may be more powerful than
(C,l). In (1.3) let p = 1, nan = - π / 2 ; the series Σ * = ι (-l)n~ ι n (that is,
un — (—l)nn) is not summable (C, 1), but summable (C, 2) to 1/4. Now
in t n ~ ( - l ) n [sin ntn + sin (n + l) tn]sin
where ntn = π—7T/2n. Hence, as τι 00
An ~ 1/4 + o(l) .
An even more striking example is un
= ("~l)n n2 .
3. Summation by harmonic polynomials. We get a more powerful method if we
introduce the harmonic polynomial
and the corresponding transform
(3.2) Bn= Σ »vPn
or
βn = tn
ϊhn(pnf tn)
Let
n
Sn = Σ
where
ON SOME TRIGONOMETRIC TRANSFORMS 2 9 5
fc _ ( k ' + l ) ••• {k + n ) nk
Ύn =n! Γ(k + 1) '
we also write
and
σ*-fLn y
ΎnNow {C,k) summability of the sequence [sn] to s is defined by
lim σ\f = s •
We quote the following elementary theorem [cf. 6, Theorem l ] , which is included
in a more general result of Mazur [ l , Theorem X] :
LEMMA 1. Let k be a given positive integer^ and let
n = 0,1,2, •••.
In order that lim Tn exist, whenever the sequence {sn} is \C9k)summable to s,
it is necessary and sufficient that:
n
<3 3 ) Σ Ύv \&antv I = 0(1) , α M = 0 f o r ^ > n ;
^ ^ lim y ί Δ α π > v = α v βΛ ί s ί s , v — 0, 1, 2,n-»oo
S) lim V απ v = /3 exists.π-»α> ^^ '
We then have lim Tn — βs + Σ^=o CXv(σv ~"-s) Since then the transform ΓΛ
296 OTTO SZASZ
is convergence preserving we must have (3.5) and:
lim anv exists,n-»co
V = 0 Ί 2" υ > J-f *>t t
hence (3.4) and (3.5) hold, so that the conditions of Lemma 1 reduce to (3.3). In
the case of the transform Bn, we have
«π,n = Pitsin nt n
sin {y + l) tn
(* + l ) t Λ '
hence
To satisfy (3.3) we must have
(3.6)
(3.7)
0 ,
sin nt
n pn
-i sin (n -\) tn
77{n - 1) tn
= 1,2,
as
and
(3.8)
k n-k s i n (n
P ~7(π -fe) tn
n-k-ls in
= 0(1)
_ sin ntn
Assume first that k = 0 then our conditions become:
(3.9)
and
(3.10) - -" S i n Vtn - S i Π {V + 1 } t n = 0(1)
ON SOME TRIGONOMETRIC TRANSFORMS 297
We now prove the lemma:
LEMMA 2. / /
(3.11) pS- ••— ρn
α s t n Φ 0 ,
then Rn is a regular transform.
Clearly (3.9) holds, and we need only to show that (3.10) also holds.
If pn > 1, then p% < p%, v - 0, 1, , n - 1 if on the other hand pn < 1,
then p% < 1. Hence, in either case,
max pi = 0(1) ,0<v<n as n
00
We have
sin vt sin (v + l) t~ P
v v + 1<
sin vt sin (v + l ) t
Σs i n
+ 1
+ l ) t
v + 1
the second term is O(t), and
sin (v + l) ts m
-f 1= Γ° cos (v = 0{t2) ,
so that
ΣPV sin vt sm
+ 1= 0
Thus (3.10) is satisfied and Lemma 2 holds.
Note that the condition p% = 0(1) is equivalent to n(pn ~ 1) < c, a positive
constant (see [5>p. 73]); furthermore, if ntn = 0(1), then clearly the secondcon-
dition of (3.11) holds.
Next let k — 1 we shall prove the theorem:
298 OTTO SZASZ
THEOREM 2. //(3.11) holds, and if
(3-12) PX s i n ntn=O(tn),
then Bn includes (C91).
The conditions (3.6)—(3.8) now become :
p5 sin ntn =O(tn) ,
pZ sin (n - 1) tn =O(tn) ,
CO
and
(3.13)n-2
Σv-l
sin vu
v= o{tn), as n • • o o
Clearly, we need only to show that (3.13) is satisfied. Now
sin vtΔ2 pι = Δ 2 / f* cos vx dx = RΔ2 f* pveivx dx
= Hfo
tpveivx(l-2peix +p2e2ix) dx
= Kft pveivx(l-peix)2 dx
Hence
sin vt
v<PV Γ \ 1 - P e ί x \ 2 d x < p v t { ( l - p ) 2 + p t 2 } ;
it follows from (3.11) that
Σsin vtn >] Σ
This proves (3.13) and Theorem 2.
4. Comparison of Bn and (C, k), k > 2. We wish to prove the following theo-
rem :
ON SOME TRIGONOMETRIC TRANSFORMS 299
THEOREM 3. Suppose that (3.11) holds and that
(4 D n*"VS sin n t B = O ( t B ) ,
(4.2) nk~ιρ$ cos ntn =0(1) ,
then Bn includes (C,k) summabilίty.
pnt Λ Φ 0 ,
Now (3.6) holds because of (4.1), and then (3.7) follows from (4.2). It remains
to prove (3.8). We have
hence
(4.3)
sin
V J*
•pvsin vt
<pvfo
t | i -/
-peίx)k+1 dx;
It follows that
(4.4)v = l
sin vtn
Pnvtn
Λ+i]
= 0 (l-Pn)k+1 Σ ^ +0 K
Here the first term is 0(1) by Lemma 2 of [ό] finally
-0(1)
300 OTTO SZASZ
This proves Theorem 3.
An interesting special case is tn = 7τ/n; the conditions now reduce to the
single condition
If, in particular, nkp% — 0(1) for all k9 then Bn includes all (C, k).
Observe that by Lemma 1 of [6] the condition n p% — 0(1) is equivalent to
lim sup \n(pn - l ) + fe log n] < +«> .
Note also that (4.1) and (4.2) imply:
n*-VS = 0(1) .
5 Truncated Riemann summability The series Σ v=0 uv is called (R9k)
summable to s if the series
(5.1)CO / . Λk
^ / s i n nt\ , N
+ Σ I — I "IE =Rk(t)n = l nt
converges in some interval 0 < t < t0, and if
Rk(t)—>s, as t •0.
For A; = 1 it is sometimes called Lebesgue summability. The method (/?, k) is
regular for k > 2 and, in fact, it is more powerful than (C9 k — 2) for k = 2, it
was employed by Riemann in the theory of trigonometric series. We generate from
it by truncation the triangular series to sequence transform {u0 — 0):
sin vtn
n - l
= Σsin vtn sin ntr.
ntr
k is a positive integer. We assume k > 2; it is then easy to show that Dn is a
regular transformation.
From Lemma 1 we find for (C9 k) to be included in Dn the conditions:
(5.2)
(5.3)
t;* (sin ίΓ^TtJ* =0(1), for v = 0,1, * , k
n-k-lsin vtn = 0(1), * oo
ON SOME TRIGONOMETRIC TRANSFORMS 3 0 1
It follows from (5.2) (see Section 2) that we must have
(5.4) ntn - pπ + 0Ln , n an = θ(l) , p a positive integer
now (5.2) reduces to
tn sin (θLn~vtn) = θ(l) , V = 0, 1, , k ,
and this is satisfied in view of (5.4).
To show that now (5.3) also holds, we employ a lemma, due to Obreschkoff
[2,p. 443]:
LEMMA 3. We have
sin vt
vt<M
v
where M is independent of t and V.
It now follows that
Σ ^sin vtn) = O(ntn) =0(1), > 0 0
This yields the following theorem:
THEOREM 4. If ntn — pu + ctn, p a positive integer^ n<Xn — 0(1), then the
transform
JL lsinvtn\k _.Λ
includes {C9k) summability (k a positive integer).
6. A converse theorem* We shall establish the following result.
THEOREM 5. //
k(6.1) lim inf
sin ntn = λ > 1/2 ,
then the transform Dn is equivalent to qonvergence.
302 OTTO szXsz
It follows from (6.1) that lim sup ntn < 2i/k hence (see Sections 1 and 5) the
transform Dn is regular. We now wish to show that Dn —> 5 implies sn —» s;
we follow a device used by R. Rado [3] •
Assume first that s = 0, and that sn — 0(1); then
0 < lim sup \sn I = δ < °o fπ-»oo
and we shall show that S = 0. To a given e > 0 choose n — n(β) so that js v | <
8 + € for v > n. Next choose m > n and such that \sm\ > δ — £. We have
sin mt.
where
Jfίtn
sin vtnin iy + 1) Vfsin
+ 1) t
hence, as mt < 77, we have
s in
/nt-
m - 1
Σ
<o(l) + (δ -h e) fsm ntnf s in mtn
nit.
It follows that
δ - β < Is, I < o ( l ) + (δ + e ) {1/λ- 1 + o ( l ) } .
But l /λ < 2, and € is arbitrarily small; hence δ = 0.
We next assume s = 0 and lim sup \sn\ — °° choose € > 0 and ω large.
Denote by m = m(α ) the least m for which | sm \ > ω; then
ω< \sm\ < o ( l ) +
But this is impossible for λ > l/2, small e, and large m.This proves our theorem
for s = 0. Finally, applying this result to the sequence \sn ~~ s |and its transform
completes the proof of Theorem 5.
7. Application to Fourier series* Suppose that f(x) is a Lebesgue integrable
ON SOME TRIGONOMETRIC TRANSFORMS 3 0 3
function of period 277, and let
GO
(7.1) f(x) ~ αo/2 + Σ (αn c o s n x "*" bn s i n nx) = Σ u n (*) ί
we may assume here α0 = 0. Now (cf. [7,p. 27])
00 ^
F(X) — f f(t) dt = C + ^ (aa sin nx ~~ 6 a cos ΠΛ) — ,0 γi
where
00 -,
c = ? „ n
It is known [7, p. 55] that at every point x where F'(x) exists and is finite, the
series (6.1) is summable (C?r), r > 1, to the value F'(x).
It now follows from Theorem 3 for k — 2 and tn — τr/n that if np\ ~ 0(1), then
" v sin vπ/n t
Furthermore, Theorem 4 yields, for k — 2, that if
ntn = prr + α n , nα n = θ(l) ,
then
n /sin vίn\2
uv \x) i i ' r V /
An analogous theorem holds for higher derivatives (cf. [7, p. 257] ).
304 OTTO SZA'SZ
REFERENCES
1. St. Mazur, Uber lineare Limitierungsverfahren, Math. Z. 28 (1928), 599-611.
2. N. Obreschkoff, Uber das Riemannsche Summierungsverfahren, Math. Z. 48 (1942-43), 441-454.
3. R. Rado', Some elementary Tauberian theorems (I), Quart. J. Math., Oxford Ser. 9(1938), 274-282.
4. W. Rogosinski, Abschnittsverhalten bei trigonometrischen und Fourierschen Reihen,Math. Z. 41 (1936), 75-136.
5. Otto Szasz, Some new summability methods with applications, Ann. of Math. 43(1942), 69-83.
6. , On some summability methods with triangular matrix, Ann. of Math. 46(1945), 567-577.
7. A. Zygmund, Trigonometrical series, Monografje Matematyczne, Warszawa-Lwow,1935.
NATIONAL BUREAU OF STANDARDS, LOS ANGELES
ON ISOMETRIC ISOMORPHISM OF GROUP ALGEBRAS
J. G. WENDEL
l Introduction. Let G be a locally compact group with right invariant Haar
measure m [29 Chapter XI]. The class L(G) of integrable functions on G forms a
Banach algebra, with norm and product defined respectively by
IWI=/U(g) !
The algebra is called real or complex according as the functions x(g) and the
scalar multipliers take real or complex values.
Suppose that T is an isomorphism (algebraic and homeomorphic) of the group G
onto a second locally compact group Γ having right invariant Haar measure μ;
let c be the constant value of the ratio m(E)/μ(τE), and let χ be a continuous
character on G. If T is the mapping of L (G) onto L (Γ) defined by
(Tx)(τg)=cχ(g) χ(g), xCL(G),
then it is easily verified that Γ is a linear map preserving products and norms;
for short, T is an isometric isomorphism of L (G) onto L (Γ).
It is the purpose of the present note to show that, conversely, any isometric
isomorphism of L (G) onto L(Γ) has the above form, in both the real and complex
cases.
We mention in passing that if T is merely required to be a topological iso-
morphism then G and Γ need not even be algebraically isomorphic. In fact, let G
and F be any two finite abelian groups each having n elements, of which k are of
order 2. Then the complex group algebras of G and Γ are topologically isomorphic
to the direct sum of n complex fields, and the real algebras are topologically iso-
morphic to the direct sum of k + 1 real fields and (n — k — l)/2 two-dimensional
algebras equivalent to the complex field. The algebraic content of this statement
Received October 24, 1950.
Pacific /. Math. 1 (1951), 305-311.
305
306 J. G. WENDEL
follows from a theorem of Perlis and Walker [4] , but for the sake of completeness
we sketch a direct proof.
Since the character group of G is isomorphic to G there are exactly k characters
Xi 9 X2 > # # " f Xk on G of order 2. Together with the identity character χ 0 these
are all of the characters on G which take only real values. The remaining charac-
ters Xk + ι> * ' •» Xrc-i f a l l i n t 0 complex-conjugate pairs, χ 2 m = χ 2 m + i> ™ =
(k + l)/2, (k + 3)/2, • • • , ( « - 2)/2. For 0 < y < n - 1 let Xj £ L (G) (complex)
be the vector with components (l/rc)χy(g). It is readily verified that the Xj are
orthogonal idempotents, so that L (G) can be written as the sum of n complex
fields, and the same holds for the complex algebra L ( Γ ) . In the real case we
retain the vectors XJ for 0 < / < k, and replace the remaining ones by the (real)
vectors ym — x2m + *2m + t 9 zm ~ iχim ~~ iχ2m + ι » whose law of multiplication is
easily seen to be yl = ym , z2
m = - y m , ymzm = zmym = z m , while all other
products vanish. Since the vectors xj, y m , 2rm span L (G) we see that L (G) is
represented as the sum of k + 1 real fields and (n — & — l)/2 complex fields; the
same representation is obtained for the real algebra L(Γ) ; this completes the
proof of the algebraic part of the assertion. The fact that these algebras are also
homeomorphic follows from the fact that all norms in a finite dimensional Banach
space are equivalent.
2. Statement of results* For any fixed g0 £ G let us denote the translation
operator x(g) —> x(golg)i x C L{G), by SgQ; operators Σγ are defined
similarly for L ( Γ ) . In this notation our precise result is:
THEOREM 1. Let T be an isometric isomorphism of the (real, complex) algebra
L {G) onto the (real, complex) algebra L (Γ). There is an isomorphism r of G onto
F, and a {real, complex) continuous character X on G such that
(1A) TSST'1 = χ ( g ) Σ τ g , g G G,
(IB)* (Γ*)(τg) = c χ(g) x(g), g G G, x CL(G) ,
where c is the constant value of the ratio m{Ej/μ(τE)
For the proof we make use of a theorem due to Kawada [3] concerning positive
*I am obliged to Professor C. E. Rickart for suggesting the probable existence of aformula of this kind.
ON ISOMETRIC ISOMORPHISM OF GROUP ALGEBRAS 3 0 7
isomorphisms of L (G) onto L(Γ) in the real case; a mapping P : L (G) —» L(Γ)
is called positive in case x(g) > 0 a.e. in G if and only if (Px)(γ) > 0 a.e. in
Γ. Kawada's result reads:
THEOREM K. Let P be a positive isomorphism of L (G) onto L (Γ), both alge-
bras real. There is an isomorphism r of G onto Γ such that PSgP~ι = kgΣTg,
g G G9 where kg is positive for each g.
In order to deduce Theorem 1 from Theorem K we need two intermediate results,
of which the first is a sharpening of Kawada's theorem, while the second reveals
the close connection which holds between isometric and positive isomorphisms.
THEOREM 2. Let P be a positive isomorphism of real L(G) onto L(T)*Then:
(2A) P is an isometry;
(2B) kg = 1 for all g C G;
(2C) P is given by the formula (Px){rg) = cx(g), where c is the constant value
of the ratio m(E)/μ(τ E)
THEOREM 3. Let T be an isometric isomorphism of L(G) onto L(V). There is
a continuous character χ(y) on Γ such that if the mapping P : L (G) —ϊ L (Γ) is
defined by {Px)(y) = χ(y){Tx)(γ), x € L(G), y C Γ, then P is a positive
isomorphism of the real subalgebra of L (G) onto the real subalgebra of L (Γ). The
character X is real or complex with L (G) and L (Γ).
3 Proof of Theorem 2. P and its inverse are both order-preserving operators,
and therefore are bounded [ l , p 249] Consequently the ratio \\Px \\ /\\x || is
bounded away from zero and infinity as x varies over L (G), x ψ- 0. If x is a posi-
tive element of L (G) it follows by repeated application of Fubini's theorem that
\\xn\\ - \x\n\ since Px is also positive, and P (xn) = (Px)n, we have the result
that for fixed positive x ^ 0 the quantity {\\Px \\/\\x \\}n is bounded above and
below for n — 1,2, . Hence P is isometric at least for the positive elements
of L (G). But now for any x C L(G) we may write x = x + x , where x and x
denote respectively the positive and negative parts of x. Then
11*11 = llχ+ + χΊI = ll«+ll +ιl*ΊI = llΛc+|| + ||Λtl>||ft+ + p«Ί| = ||fte||.
Applying the argument to P~ι we obtain the result
11*11 " I I
308 J. G. WENDEL
which is the statement (2A).
Theorem (2B) follows at once from this and Theorem K. For if x £ L (G) then
IISgX II = TOg||#||, where πig is the constant value of the ratio m(gE)/m(E). Simi-
larly, | | Σ T g ^ : | | = μTg \\ζ\\ Since r is a homeomorphism,μ τg — mg.The constant
kg may now be evaluated by taking norms on both sides of the equation PSgP~ι
= kgΣTg, and must therefore have the value unity.
To prove part (2C) of the theorem we observe that the operator Q defined by
(Qx)(τg) — cx(g) satisfies the relation QSgQ~x — Σ T g, and is an isomorphism of
L (G) onto L (Γ) . Then QSgQ~ι = PSgP~ι, g £ G, and consequently R = P~ιQ
is a continuous automorphism of L (G) which commutes with every Sg. We shall
show that R must be the identity mapping.
Segal [5, p. 84] has shown that the product xy of two elements x9 y belonging
to L (G) may be written as a Bochner integral, which in our notation takes the form
xy= Jx(h)mlι{Shy}m(dh),
where the quantity in braces is a vector-valued function of h £ G9 and the function
mg was defined above. Applying the operator R we obtain
R(xy) = $x{h)*iι{RShy\m(dh) = SxWm^iS^ylmidh) =xRy.
But R is an automorphism, and so also R (xy) — (Rx)(Ry). Thus x = Rx, all
x G L (G), which shows that P — Q9 as was to be proved.
4 Proof of Theorem 3 We first require several lemmas, all of which share the
hypothesis: T is an isometric isomorphism of L (G) onto L ( Γ ) , indifferently real
or complex. For x, γ £ L (G) we write ξ for Tx, 7) for Ty. We denote by E (x) thes e t £g|& €1 G9x(g) 7^ θ}9 which is regarded as being determined only up to a
null-set; E (ζ) in Γ is defined in the same fashion. (Although we make no use of
this fact, the first three lemmas below actually hold in case T is an isometry
between two arbitrary L-spaces.)
LEMMA 1. If EM Π E (y) = Λ then E {ξ) Π E (17) = Λ , and conversely.
Proof. The hypotheses imply that for all scalars A we have \x + Ay\\ — \\x\
+ \A\ \\y\\. Then for all A we have \\ξ + AΎ)\\ = \\ξ\\ + \A\ | | η | | , which implies
that E {ξ) and E (η) are disjoint. For the converse we need only replace T by T *.
ON ISOMETRIC ISOMORPHISM OF GROUP ALGEBRAS 3 0 9
LEMMA 2. IfE(x) C £ (y) then E(ξ) C £ (η), and conversely.
Proof. Suppose that E (x) C E (y), but that E (ξ) $ E(η). Then we may
write ξ= ξx + ξ29 w i t h f i ^ ) C £(η), £ ( £ 2 ) Π £ ( η ) = Λ = E (ξ x) Π E (ξ2).
Let Γ x ^ = xι\ then from Lemma 1 it follows that E(xχ) Π E(x2) — A = £Gt2)
Π £(y). But E{xχ) U E(x2) = E (x) C £ (y); this contradiction yields the result.
LEMMA 3. Let B in Γ be a σ-finite measurable set {that is, the sum of a
countable number of sets of finite measure). Then there is a positive x C L (G)
such that E {ξ) - B.
Proof. Let 77 £ L{Γ) be chosen so that £ (η) = B. Let y = T~ιη, and setΛ Q>) = ITQ>) I > £ £ G. Then Λ C L (G)> E{X) — E (y), and therefore from Lemma
2 it follows that £ (£) = β.
LEMMA 4. Le£ A; o/icί y be positive elements of L(G). For y C E{ξ) let
Kξ (γ) = ξ(y)/\ζ{y)\9 and define Kv (y) in similar fashion. Then Kξ (y) =
Kv (γ) almost everywhere on E (ξ) Γ) E (η).
Proof. Since x and y were taken to be positive we have \\x + y | |= ||Λ;|| + ||y||.
Therefore \\ξ + η | | = | ^ | | + |[^H. Then \ξ(γ) + η(Ύ)\ = \ξ(γ)\ + \V(γ)\
a.e. in F . Hence, since the functions K have modulus 1,
\κξ(y)Kη(γΓ1\ξ(y)\ + \V(Ύ)\\ = +a.e. in £ (ξ) Π E (η). But then Kξ {y)Kv (y ) " 1 = 1 a.e. on £ (£) Π £ (η), as was
to be proved.
LEMMA 5. There is a unique continuous character χ on Γ with the property
that for all positive x G L(G) we have ξ{y) = X ( y ) | £ ( y ) | a.e.\ χ is reaZ or
complex with L (G) and L (Γ).
Proof. Let Fo be the open-closed invariant subgroup of Γ generated by a
compact neighborhood of the identity. Since Γo is σ-finite we may apply Lemma 3
to obtain a positive x £ L{G) such that £ (ξ) — Γo . Now x > 0 implies that
| |* 2 I = I* I 2 ; then also ||<f 2 ' | | = ||<f | | 2 . The element ξ2 is given by the formula
Since * 2 is also positive we have from Lemma 4 that X^ 2 (y) = ^ ^ (y) a e o n
E(ξ2) Γ\ E(ξ) C Fo = E{ξ). Writing simply K{y) for the common value, we see
310 J. G. WENDEL
that the relation ξ2{y) = K(γ)\ξ2(γ)\ therefore holds in Γo even outside ofE (ξ2). Then
\ξ2{y)\ = JTo
Integrating over Γo again we obtain
= U\\2 = hidy) J
Therefore K(γ)~ιK(γδ~ι)K(8) = 1 a.e. on Γo X Γo. Then there is a null-set
i V C Γ 0 such that y €[ N implies K (yS"1) K(8) = K(γ) for almost all 8 G Γo.
We integrate this equation over a set M of finite positive measure and obtain
K(Ύ)μ(M) = JVo
φu(Sγ)μ(d8),
where φy is the characteristic function of M. The right member is easily seen to
be a continuous function of 7, for all 7 £ Γo hence £ ( 7 ) is equal a e. to a
continuous function Xo(7)> which is clearly a character on Γo. From Lemma 4 it
follows also that, for positive x £ L (G), if E (ξ) 9 T o then ξ(γ) = χ 0 (7)
The proof is completed by extending the function χ 0 to all of Γ. To do this we
write Γ as the union of disjoint cosets 7αΠ)> a n ^ consider the open-closed sub-
group Γ\ generated by any finite number of cosets. Then Γ\ is again σ-finite, and
we may repeat the above argument to obtain a continuous character )(i on Fj
Lemma 4 guarantees that for two such subgroups Γ\ and Γλ the characters )( t and
Xi will agree on Γx Π Γ/ 2 Π)> s o ^at Xi ^s indeed an extension of χ 0 . Clearly,
i f * > 0 a n d £ ( £ ) C Γt then ^(7) = χ ^ γ ) | ^(7) | .
Finally, X on all of Γ is defined by χ(y) = χ x (y) for y £ Γi Since the union
of all such subgroups ΓΊ is precisely Γ, and since as shown above the subgroup
ON ISOMETRIC ISOMORPHISM OF GROUP ALGEBRAS 3 1 1
characters are mutually consistent, the function X is well-defined. It is clearly a
continuous character. The remaining property, that x > 0 implies ζ{y) = χ ( y )
l ί ί y ) ' * can be proved as follows. The set E (ζ) intersects at most a countable
number of cosets ynΠ> ^n s e t s °f positive measure. Let ξn be the restriction
to ynV0 of ξ, and put xn = T~ιξn. Then x = Σn=ιXn, and by Lemma 1 the sets
E{xn) are pairwise disjoint, so that the xn are themselves positive elements. From
this it follows that^(y) = χ Λ ( y > | £ n ( y > | = χ(y)\ξn(y)\ f o r r G r Λ r 0 ;
hence the result holds.
The proof of Theorem 3 is now immediate. For the continuous character X on
Γ constructed in Lemma 5 the mapping P on L (G) to L (Γ) defined by (JPx){γ)
= χ(y)~ι (Tx) (y) carries positive elements of L (G) into positive elements of
L (Γ); P is clearly an algebraic isomorphism of L (G) onto L (Γ). We have only to
show that Px positive implies x positive. Suppose then that Px = ξ is positive,
but that x = xx ~ x2 + i (x3 - * 4 ) , with XJ > 0 and E (xx) Π £ 0c2) = £ (x3) Π £ ( * 4 )
= Λ, and correspondingly ξ = ξγ ~ ^ 2 " ι (^3 "~ ^4)* ^s evidently an isometry,
and therefore by Lemma 1 the sets E(ξι) Π £(^2) a n ( l E(<ξ3) Π E(ξΛ) are null-sets.
Therefore ^ 2 = ^ 3 = ^ 4 = 0 ; so x ~ Xι 9 and x is positive.
5 Proof of Theorem 1. Because of Theorem 3 we may apply Theorems Kand
(2B) to the real sub-algebras of L(G), L (Γ), to conclude that there is an iso-
morphism T of G onto Γ such that PSgP~ι — Σ T g . Since r is a homeomorphism we
may regard the function X as a continuous character on G, by defining χ(g) =
χ(rg). By Theorem (2C), P is given on the real subalgebras by the formula (Px)
(rg) = ex (g), and, because of the linearity, this formula must hold throughout all
of L (G). Therefore (Tx)(rg) = cχ(g) x{g), which proves (IB). Theorem (1A) is an
easy consequence of this formula.
We note finally that Theorem (2A) shows that Kawada's theorem follows from
Theorem 1.
REFERENCES
1. Garrett Birkhoff, Lattice Theory, Amer. Math. Soc. Colloquium Publications, vol.25;American Mathematical Society, New York, 1948.
2. P. R. Halmos, Measure Theory, D. Van Nostrand, New York, 1949.
3. Y. Kawada, On the group ring of a topologicaί group, Math. Japonicae 1 (1948), 1-5.
4. S. Perlis and G. L. Walker, Abelian group algebras of finite order, Trans. Amer.Math. Soc. 68 (1950), 420-426.
5. I. E. Segal, Irreducible representations of operator algebras, Bull. Amer. Math. Soc.53 (1947), 73-88.
YALE UNIVERSITY
ON THE Lp THEORY OF HANKEL TRANSFORMS
G. M. W I N G
l Introduction. Under suitable restrictions on f(x) and vy the Hankel trans-
form g(t) of f(x) is defined by the relation
ω g ( t ) = $* {χ
The inverse is then given formally by
(2) / ( * ) = J f (xtY/2Jv(xt)g(t)dt.
These integrals represent generalizations of the Fourier sine and cosine trans-
forms to which they reduce when V = i: 1/2. The L p theory for the Fourier case
has been studied in considerable detail. In this note we present some results con-
cerning the inversion formula (2) in the Lp
a case.
It is clear that if f(x) £ L and H(v) > —1/2 then the integral in (1) exists.
It has been shown [3,6] that if f{x) ζlLp,l<p<2, then
converges strongly to a function g (t) in L p . For this case Kober has obtained the
inversion formula,
/(«) == ..-1/2-V A.
dxv+1/2 ί
«, (xt)1/2 Jv+t(xt)
which holds for almost all x In her investigation of Watson transforms, Busbridge
[ l ] has given analogous results for more general kernels. Except when p = 2
the question of the strong convergence of the inversion integral has apparently
been considered only in the Fourier case [2] . We now investigate this problem
Received January 10, 1951.
Pacific J. Math. 1 (1951), 313-319.
313
3 1 4 G. M. WING
for the Hankel transforms. We assume throughout that R(v) > —1/2.
2. Theorem. We shall establish the following result.
THEOREM 1. Let fix) G Lp
9 1 < p < 2 , and let git) be the limit in
mean of gait), git) = l.i.m. gait), where gait) is defined by (3). //
/ . G O - so
a (χtr>jv{χt)g{t)dt,
then
fa(x) G Lp and /(*) = Li.m. fa (x) .
Proo/. Write
/.(* .* )= / o
α (χt)1/2
(xu)1/2f(u)dufo
aJv(ut)jv(xt)tdt.
Since gbit) converges in the mean to git) it follows that lim^o, faix$b) = faiχ)
Hence
where [9]
(5) ίT(x,ufα)= Jfα Jv{ut)Jv(xt)tdt
An integral very similar to (4) has been studied in a previous paper [lO] . The
same methods may be used here to show that || /a(*)||p < Λίp|| /(#)||p Our theorem
will now follow in the usual way if we can prove it for step functions which vanish
outside a finite interval. Let φix) be a step function, φix) = 0 for x > A, and
let φaix) correspond to it as in (4). Choose ξ > 2A9 a > A, to get
Jg* \Φa(x) -Φ{x)\pdx = •// dx )// φ(u)(χUy* K(x,u,a) da \".
ON THE Lp THEORY OF HANKEL TRANSFORMS
From the relations
(6) xin Jv{x) = (2/π)1/2 {cos (* + K) + x~lAv sin (* + K)\ + θ{x~2)
(«-•»)..
where
Av = (l - 4v2)/8 , K =-(2v+ l )τr/4,
and
(7) Jv{x) = O(xvi) (* —» 0 ) ,
where 1^ = R (v), it is easy to see that
so that we have
/ / \Φa(x) -φ
for ξ sufficiently large. Now
As α —» 00 the integral goes to zero by the L2 theory for Hankel transforms
(see [7, Chapter 8J ). This completes the proof.
3 The case p — 1. Theorem 1 fails to hold in the case p = 1. The proof,
similar to that given by Hille and Tamarkin in the Fourier case [2] , will only be
sketched*
THEOREM 2. There exists a function h(t), the Hankel transform of a function
φ{x) £ Lysuch that if
(8) Ψa(x)= fo
a ( * 0 1 / 2 Ju(xt)h(t) dt
then l.i.m. ψa(x) fails to exist.
3 1 6 G. M. WING
Proof. Let h(t) = tί/2 /v(ί)/log(ί + 2). Two integrations of (8) by parts and
use of formulas (5), (6), and (7) yield
( 9 ) ^ α W ( 2 Λ\ Ί
U^-l) logfor large Λ; .
Now define ι//(#) = limα-co ψaix). It is evident from (8) that i/>(x) is con-
tinuous except perhaps at Λ; = 1, while (9) shows that ψix) = 0 ( # " 2 ) . To show
that ι//(#) C L it suffices to consider the neighborhood of x = 1. Formula (6)
yields, after some calculation,
/ \ /•<» cos (1 — x) t , xΨ W = ί log (, + 2) " ' + α ( ' ' '
where <χix) is continuous near x = 1. Thus
dt+ fW 5 J° ί log (2 + t/β) J o t log (2 + t)
The first integral on the right tends to zero as € —» 0 . Since ψix) ~" CC(%) is
positive (see [2] ) it follows that ψix) — OC(Λ ) is integrable over (1,2) [β,
p. 342] . The interval (0,1) may be handled similarly. Hence ψ(x) C L .
That hit) is indeed the Hankel transform of ψix) is a consequence of a result
of P. M. Owen [5,p.31θ] . But it may be seen from (9) that ψaix) is not in L , so
that l.i.m ψaix) surely fails to exist.
4. A summability method. It is natural to try to include the case p = 1 into
the theory by introducing a suitable summability method. Our interest will be con-
fined to the Cesaro method. If fix) £ L and git) is its Hankel transform then we
shall define
/.(*) = fo° (1 - t/a)k{xt)U* Jv(*t)g(t) dt
= C f(y)ck(χ,y,a) dy,
ON THE Lp THEORY OF HANKEL TRANSFORMS 317
where
(11) Ck(x,y, a) = f* (xy)1/2 uJ v(*α)j v(yu)(l - u/a)k da.
Offord [4] has studied the local convergence properties of fa(x) for k = l.We
are able to extend his results to the case k > 0, but the estimates required are
too long and tedious for presentation here. Instead we investigate the strong con-
vergence.
THEOREM 3. Let fix) G L, k > 0. // faix) is defined by (10), then faix)
converges strongly to fix).
Proof. We shall first prove that Cjcix9y9a) C L and ||C&(#>y,α)|| < M,
where the norm is taken with respect to x and the bound M is independent of γ and
a. An integration by parts and a change of variable in (11) give
/Ί o) C ( \ — I (Λ — λκ~"l ( \l/2 Π J
2 ^
where
= Jy+i(ays)Jv(axs) - Jv
y - x
Jv+ι{ays)jy{axs) + Jy{ays)jv+ι{axs)*~ .
Consider
where
Jv+1(ays) (θ < s < l),
(s > 1).
318 G. M. WING
Now, as a function of s , G(a9y9s) £ Lp for some p > 1 so that
F(a,y,z) = fΰ
ωG(a,y,s)(sz)1/2Jv{sz)ds
is in Lp as a function of z [3] Also
a'<Ap JjΓ" \G(a,y,s)\Pdsy/P <M,
where M is a constant independent of a and y. Thus
The other parts of (12) may be cared for similarly, so that we have
The range | y — x \ < I/a is easily handled since, by (11), for this range we have
y,α) | < Ma- Hence | |C^(x,y,a)| | < M. We see at once from (10) that
C \fa(χ)\dx= tfdx
< C \f(y)\dy / / \Ck(x,y,a)\dx,
so ||/a(^)|| ^ ^l l/(*) | | The proof may now be completed by the methods of
Theorem l
REFERENCES
l I. W Busbridge, A theory of general transforms for functions of the class Lp(0,«>)(1 < p < 2), Quart. J. Math., Oxford Ser. 9 (1938), 148-160.
2. E. Hille and J. D. Tamarkin, On the theory of Fourier transforms, Bull. Amer. Math.Soc. 39 (1933), 768-774.
3. H. Kober, Hankelsche Trans format ionen, Quart. J. Math., Oxford Ser. 8 (1937),186-199.
4. A. C. Offord,0τι Hankel transforms, Proc . London Math. Soc. (2) 39 (1935), 49-67.
ON THE Lp THEORY OF HANKEL TRANSFORMS 319
5. P. M. Owen, The Riemannian theory of Hankel transforms, Proc. London Math. Soc.(2) 39 (1935), 295-320.
6. E. C. Titchmarsh, A note on Hankel transforms, J. London Math* Soc. 1 (1926),195-196.
7. , Introduction to the theory of Fourier integrals, University Press, Oxford,1937.
8. r The theory of functions, University Press, Oxford, 1932.
9. G. N. Watson, Theory of Bessel functions, University Press, Cambridge, England,1922.
10. G. M. Wing, The mean convergence of orthogonal series, Amer. J. Math. 72 (1950),792-807.
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Pacific Journal of MathematicsVol. 1, No. 2 December, 1951
Tom M. (Mike) Apostol, On the Lerch zeta function . . . . . . . . . . . . . . . . . . . . . . 161Ross A. Beaumont and Herbert S. Zuckerman, A characterization of the
subgroups of the additive rationals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169Richard Bellman and Theodore Edward Harris, Recurrence times for the
Ehrenfest model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179Stephen P.L. Diliberto and Ernst Gabor Straus, On the approximation of a
function of several variables by the sum of functions of fewervariables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Isidore Isaac Hirschman, Jr. and D. V. Widder, Convolution transforms withcomplex kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Irving Kaplansky, A theorem on rings of operators . . . . . . . . . . . . . . . . . . . . . . . 227W. Karush, An iterative method for finding characteristic vectors of a
symmetric matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233Henry B. Mann, On the number of integers in the sum of two sets of positive
integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249William H. Mills, A theorem on the representation theory of Jordan
algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255Tibor Radó, An approach to singular homology theory . . . . . . . . . . . . . . . . . . . . 265Otto Szász, On some trigonometric transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 291James G. Wendel, On isometric isomorphism of group algebras . . . . . . . . . . . 305George Milton Wing, On the L p theory of Hankel transforms . . . . . . . . . . . . . 313
PacificJournalofM
athematics
1951Vol.1,N
o.2
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