Pacific Journal of Mathematics IN THIS ISSUE— Tom M. (Mike) Apostol, On the Lerch zeta function ............... 161 Ross A. Beaumont and Herbert S. Zuckerman, A characterization of the subgroups of the additive rationals ....................... 169 Richard Bellman and Theodore Edward Harris, Recurrence times for the Ehrenfest model ........................................ 179 Stephen P.L. Diliberto and Ernst Gabor Straus, On the approximation of a function of several variables by the sum of functions of fewer variables ................................ 195 Isidore Isaac Hirschman, Jr. and D. V. Widder, Convolution transforms with complex kernels ............................ 211 Irving Kaplansky, A theorem on rings of operators ................ 227 W. Karush, An iterative method for finding characteristic vectors of a symmetric matrix ........................................ 233 Henry B. Mann, On the number of integers in the sum of two sets of positive integers ........................................... 249 William H. Mills, A theorem on the representation theory of Jordan algebras .................................................. 255 Tibor Radó, An approach to singular homology theory ............. 265 Otto Szász, On some trigonometric transforms .................... 291 James G. Wendel, On isometric isomorphism of group algebras .... 305 George Milton Wing, On the L p theory of Hankel transforms ....... 313 Vol. 1, No. 2 December, 1951
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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
E. F. BECKENBACH, Managing EditorUniversity of CaliforniaLos Angeles 24, California
ASSOCIATE EDITORS
R. P. DILWORTH
HERBERT FEDERER
MARSHALL HALL
P. R. HALMOS
HEINZ HOPF
R. D. JAMES
B0RGE JESSEN
PAUL LEVY
GEORGE POLYA
J. J. STOKER
E. G.STRAUS
KOSAKU YOSIDA
SPONSORS
UNIVERSITY OF BRITISH COLUMBIA
CALIFORNIA INSTITUTE OF TECHNOLOGY
UNIVERSITY OF CALIFORNIA, BERKELEY
UNIVERSITY OF CALIFORNIA, DAVIS
UNIVERSITY OF CALIFORNIA, LOS ANGELES
UNIVERSITY OF CALIFORNIA, SANTA BARBARA
OREGON STATE COLLEGE
UNIVERSITY OF OREGON
UNIVERSITY OF SOUTHERN CALIFORNIA
STANFORD UNIVERSITY
WASHINGTON STATE COLLEGE
UNIVERSITY OF WASHINGTON• • •
AMERICAN MATHEMATICAL SOCIETY
NATIONAL BUREAU OF STANDARDS,
INSTITUTE FOR NUMERICAL ANALYSIS
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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
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
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
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
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.
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:
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
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||.
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.
UNIVERSITY OF CALIFORNIA, LOS ANGELES
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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ASSOCIATE EDITORS
R. P. DILWORTH
HERBERT FEDERER
MARSHALL HALL
P. R. HALMOS
HEINZ HOPF
R. D. JAMES
B0RGE JESSEN
PAUL LEVY
GEORGE POLYA
J. J. STOKER
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KOSAKU YOSIDA
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INSTITUTE FOR NUMERICAL ANALYSIS
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UNIVERSITY OF CALIFORNIA PRESS BERKELEY AND LOS ANGELES
COPYRIGHT 1951 BY PACIFIC JOURNAL OF MATHEMATICS
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
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