All S7 WIENER'S APPROXIMATION THEOREM FOR LOCALLY COMPACT ABELIAN GROUPS THESIS Presented to the Graduate Council of the North Texas State University in Partial Fulfillment of the Requirements For the Degree of MASTER OF SCIENCE By Ven-shion Shu, B. 5. Denton, Texas August, 1974
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AllS7
WIENER'S APPROXIMATION THEOREM
FOR LOCALLY COMPACT
ABELIAN GROUPS
THESIS
Presented to the Graduate Council of the
North Texas State University in Partial
Fulfillment of the Requirements
For the Degree of
MASTER OF SCIENCE
By
Ven-shion Shu, B. 5.
Denton, Texas
August, 1974
Shu, Ven-Shion, Wiener's Approximation Theorem for
Locally Compact Abelian Groups. Master of Science (Mathe-
Next, suppose hcEL1 (Rn). Then there is a gce cOO(Rn) such
that ||h-g|| 1 < . For this g, we can find a finite sum
k Lakf such that ||g*f - kLkLak fil< E. Hence,k k k k
h*fD-Z \Ak Lfa 1l 1511(h-g)*f|1|,+|g*f - \A L f||k k k k ak
5 ||h-g||i- + E
< e.
(iv) Finally, let g E L(Rn) be given. Choose g0 EL1(R n)
such that go E Coo(Rn) and I|g-g 0 1 < . Let h E L (Rn) such0 2that go = h*f, and find a finite sum Z kLakf such that
fh*f - kLf 11< L. It follows then that
0 fjgk Lak1-ggj| +Jflh*f - Z)kLaf
1.13. Definition. A linear subspace I of Ll(Rn) is
said to be invariant under translation, or simply invariant,
if f E I implies Laf EI for all a e Rn.
1.14. Theorem. The closed invariant subspaces of
L (R n)coincide with the closed ideals.
Proof. 1.12 (iii) in fact says that every closed in-
variant subspace of L1(Rn) is an ideal. We prove now that
every closed ideal of L (Rn) is an invariant subspace. Let I
be a closed ideal. Let f EI and acERn be given. Given E> 0,by 1.7, there is a Gc7L1(Rn) such that |la*f..fi||< e. Since
(Laa)*f = La( a*f) = a *(Laf), therefore
1* -f f =111 ILa(a*f) -L 1f
!(Laa)*f - Laf 1
< E.
Since Lao * f FI and E is arbitrary, hence Laf E I.
14
1.15. Theorem. Let S c L (Rn) If (f:f S has no
common zero, then the closed invariant linear subspace spann
by S is L1(Rn) itself, and the converse is true, also.
Proof. Let I be the closed invariant subspace spanned
by S. Then by 1.14, I is a closed ideal. For every compact
set K, there are finitely many functions (fklskiN in I such
that at every point of K at least one of the f'5s is nonzeroN
ed
there. Let f = Z)fk*fk, where f(x) = f(-x). ThenN k=1
f k= Ifk|. Hence, f eI and f?(t) > 0 for all tcE:K. Sup-k=l
pose is a function in L (Rn) with g vanishing outside K.
Then by the Wiener-Levy theorem, there exists $ E ,(Rn) such
that i(t) = 1 for all t eK. It follows that goI
since g0 (t) = (g0*h*f5 (t) for all tE Rn. Thus, I contains
the dense set of all functions in L (Rn) whose Fourier trans-
forms have compact support. Therefore I = Ll(Rn), since I
is closed. The converse part is similar to 1.12 (A).
CHAPTER BIBLIOGRAPHY
1. Rudin, Walter, Real and Complex Analysis, New YorkMcGraw-Hill Book Company, Inc., l96X
2. Stein, Elias, M., and Guido Weiss, Introduction toFourier Analysis on Euclidean spaces, Princeton, N. J.,Princeton University Press, 1971,
15
CHAPTER II
FUNCTION ALGEBRAS AND GENERALIZATION
OF WIENER'S THEOREM
Throughout this chapter, X will always denote a locally
compact Hausdorff space, and i(X) will denote an algebra of
complex-valued continuous functions on X with the ordinary
pointwise algebraic operations.
2.1. Definition. (X) is called a standard function
algebra, or standard algebra for short, if it has the fol-
lowing two properties.
(i) If f c 6(X) and f(a) # 0 at a point a EX, then there
is a g E (X) such that g(x) =f() for all x in some neighbor-
hood of a.
(ii) For any closed set E cX and every point aEX - E,
there is an f E a(X) vanishing on E and f(a) / 0.
Remark. 7 (Rn) is a standard function algebra. This
follows from 1.~3 and 1.10, with A(z) = }1. 1(Rn) will
serve as a model of our development in this chapter.
2.2. Theorem. Let a(X) be a standard function algebra.
For any a c X and any neighborhood U a of a, there is a function
Ta Ea(X) such that Ta is 1 near a, and supp Ta C Ua
16
17
Proof. Since X is locally compact, there is an open
neighborhood Va of a with compact closure such that
a c Va CV acUa. By 2.1 (ii), there is an f E 6(X) such that
f(a) / 0 and f(vc) =0. By 2.1 (i), then, there is a g c(X)
such that g(x) = near a. Define Ta = f - gc7(X). Then
Ta is 1 near a and supp Tac Vac Ua
2._1. Theorem. Let a(X) be a standard algebra. If K
is a compact set in X and U is a neighborhood of K, then
there is a function f in c(X) such that f(K) = 1, and
supp f C U.
Proof. For each a E K there is, by 2.2, a Ta c 6(X) and
an open neighborhood Va of a such that Ta(x) = 1 for all
X E Va, and supp T a CU. (Va acK is an open covering of K.
Therefore there are finite points (ak)1l kN of K such that
N
K c k= ak Let us denote Tk Vk for Tak.9Va , 1<kSN,k~l kk k
respectively. Now define el = T1, and (if N2) e2 = T2 (-T),,*.,
eN TN(l TN-1)''(l Tl). Then, (i) ek c (X), lk5N; and
N N(ii) ZDek 1 (1- Tk), as can be shown by simple induc-
k=l k=l
Ntion. Define f = Zek E 6(X). If x E K, then x E Vk for some k,
k=1
hence Tk(x) = 1. It follows that f(x) = 1 for all x cK.
Since supp ek C U, l k N, therefore, supp f C U.
2.4. Definition. Let I be an ideal in a function
algebra c(X) and g be a (complex-valued) function on X. We
18
say that g belongs locally to I at the point aE X if there
is a function fa c I coinciding with g near a. If X is not
compact, we say that g belongs locally to I at infinity if
there is an f E I coinciding with g outside some compact set.
2.5. Theorem. Let 6(X) be a standard algebra and I be
an ideal of 67(X). Suppose g is a function on X belonging
locally to I at every point of X, and also at infinity if X
is not compact. Then g e I.
Proof. There is an f0 c I coinciding with g outside some
compact set K cX (if X is compact, let f0 = 0 and K = X).
For every a E K there is an fa c I coinciding with g in a neigh-
borhood Ua of a. There is also a Ta c 6(X), by 2.2, such that
Ta is 1 in another neighborhood Va of a, and supp T ac U.
Since K is compact, there are finite points (ak)l k<N of K
Nsuch that Kck=V ak. Let Vk'fk, Tk stand for Vakf k Tak
respectively. Now define e1 = T1 , and (if N 2)
e2 = 2~(1-Tl)5''e'N = TN(l~ TNl).(l1Tl). Then, as in 2.~5,
Neach ek c(X) and Z ek is 1 on K. Consider the function
k=l
N Nh = (1 - k= e)f +k k k E I. Notice that efk = ekg, since
k=l k=1ek(x) / 0 implies X EUk, and therefore fk(x) = g(x). Sim-
N Nilarly, (1 - Z ek)f (1 - Ze)g. Hence,
k=l k=1N N
h = (1 - Z ek)g + ( Z ek)g = g E I.k=l k=l
19
2.6. Theorem. If c(X) is a standard function algebra
and f E :(X) is non-zero on a compact set K c X, then there isa g E 6(X) such that g(x) = 1 for all x EK.
Proof. For each a c K, there is an open neighborhood U aof a such that f(x) / 0 for all xc U. Since X is locally
compact, there is an open Va having compact closure such
that aE Va c Va CUa. There are finite points (a k)l k5NsuchN Nk k
that K ck= Vak. Let K '= U V . Then K c K', K'is compact,k=l k
and f (x) / 0 for all x E K'. By 2.3, there is an f1 c(X)which is 1 on K and supp f1 c K'. Define g by g(x) - f(x) iff(x) / 0, g(x) = 0 if f(x) = 0. Therefore, if xfEK'c )ng(x) = 0. If ac K', then f(a) / 0, and there exists an
h c a(X) such that h(x) = 1 in some neighborhood of a.
Hence g(x) f(x) 1 (x) . h(x) in some neighborhood of a.
Therefore, g belongs locally to d(X) at every point of X andat infinity. The proof is then completed by 2.5 with I = c7(X).
2.7. Definition. Let fcE6I(X). The set of all zeros off is called the cospectrum of f, denoted by cosp f. Let S
be a subset in c7(X). The set of points of X where all func-tions in S vanish is called the cospectrum of S.. We denote
it by cosp S.
2.8. Theorem. If I is an ideal in a standard function
algebra 6(X), then a function f E a(X) belongs locally to I
at every point in X-cosp I.
20
Proof. If ac X-cosp I, then there is an hE I such
that h(a) ' 0. By 2.1 (i) , there is an h1 1 C6(X) such that
h() =hx in some neighborhood Ua of a. Then hh f eI,
and h h1f coincides with f on Uae
2.9. Theorem. An ideal I in a standard function
algebra 6(X) contains every function in 6(X) that has compact
support disjoint from cosp I. In particular, if cosp I = 0,
I contains all functions in a(x) A 000(X).
Proof. If f E (X) A 000(X) having compact support K
disjoint from cosp I, then by 2.8, f belongs locally to I at
every point of X and at infinity. It follows by 2.5 that
f G I.
2.10. Theorem. Let 67(X) be a standard algebra and let
E be a closed set in X. The ideal of all functions in 7(X)
with compact supports disjoint from E is the smallest ideal
of 67(X) with cospectrum E.
Proof. In view of 2.9, we need only to show that I =
IE = (f If c (X) n c00 (X) and supp f n E = 0} is an ideal,
and cosp IE = E. Since supp (f + g) c supp f U supp g, and
supp f - g c supp f n supp g, therefore, if f,g E I, c c C, and
hcE7(X), then f +gc-I, cf EI, and hf E I. This proves that
I is an idealof a(X). If ac E and fcEIE, then a/ supp f
so that f(a) = 0. Hence, Ec cosp IE. Suppose a / E,, then
there is an open set Va having compact closure such that
21
a ev CV a c Ec. By 2.2, there is a TaCE 6(X) such that Ta is 1
near a and supp Ta cV a. Therefore, Ta CIE and a /cosp IE'We have proved that cosp IE = E.
2.11. Definition. Let c(X) be a function algebra and
suppose that c(X) is endowed with a certain topology T. Then
6(X) is called a topological function algebra (with respect
to T) if it has the following properties.
(i) 6(X) is a topological vector space.
(ii) The map (f,g) - fg from a(X) x a(X) into 6(X) is
continuous in f for each fixed g.
(iii) For each acEX, the linear functional f-4 f(a) on
a(X) is continuous.
2.12. Definition. a(X) is said to be a normed function
algebra if the following holds:
(i) 6(X) is a normed vector space, with norm J| J|;(ii) |Jfg |J : ||f ||||1 g||;
(iii) for each a EX, the linear functional f-+ f(a) on
c7(X) is continuous.
It is clear that every normed function algebra is a
topological function algebra.
2.13. Definition. A topological function algebra and
a normed function algebra are called topological standard
algebra and normed standard algebra, respectively, if they
have the properties 2.1 (i), (ii).
22
2.14. Theorem. Let 7(X) be a topological standard
algebra. Let E be a closed set and IE E(Xf I f Ea(x) n c00(X)and supp ffnlE = 0). Consider J = IE. Then,
(I) JE is a closed ideal,
(ii) cosp JE = cosp IE = E,
(iii) JE is the smallest closed ideal with cospectrum E.
Proof. (i) Let gGJE = E, and fc 67(X). We need to
show that fg c JE. Let fg } be a net in IE converging to g.
Then, by 2.11 (ii)(, fg } converges to fg. Since fg aeiE9therefore fg c IE =E'
(ii) Clearly, cosp JE C cosp IE* If a /cosp JE, then
there is an f e JE such that f(a) / 0. It follows that there
must be some g E:IE such that g(a) / 0. For there exists a
net (g } in IE converging to f, hence, by 2.11 (iii),
g (a) -+ f(a). If g (a) = 0 for all a, then f(a) = 0, a
contradiction.
(iii) If I' is any closed ideal with cospectrum E,then IE c I', by 2.10. Therefore, JE EB c ' =C',
2.15. Theorem. In a topological standard algebra 67(X),the closure of the ideal of all functions in 67(X) with com-
pact supports is the smallest closed ideal with empty co-
spectrum.
Proof. This is simply an application of 2.14 with E =0.
2.16. Definition. A topological standard algebra 67(X)is called a Wiener algebra if 6(X) n Coo(X) is dense in 67(X).
235
2.17. Theorem. In a Wiener algebra i(X), a closed ideal
I coincides with 6(X) if and only if cosp I = s.Proof. Let I be any closed ideal in c(X) with empty
cospectrum. Then c(X) nc0 0 (X) c I. Therefore, I = I = a(X)Conversely, if I = a(X), then cosp I = cosp 7(x) = , by
2.1 (ii).
Remark. 2.17 is the abstract form of Wiener's approxi-
mation theorem.
2.18. Definition. We say that a topological function
algebra 6(X) possesses approximate units if for every f c a(x)and every neighborhood V 0 of 0 in a(X) there is a T c 6(X)
such that TfG f +U(0
2,19. Theorem. Let Q(X) be a topological function
algebra possessing approximate units. If a(X) A C00 (X) is
dense in c(X), then 67(X) also possesses approximate units
with compact support.
Proof. Let f E 7(X) and W be any neighborhood of 0 in
67(X). Let 1 ' be another neighborhood of 0 in 67(X) such that
0 + 0c 40 . Choose ucy c(X) so that af E f+7%' . Since
0 - f =0 c C', there is a neighborhood 'Y of 0 in 67(X) such
that 0- f Choose r c a(x) n0( such that T C 9 + V0r0 ?/0 T6(XA 0 0 (X) 0Then Tf = af + ( T-a)f C f + + - f c f+ +%c f +('U
2.20. Theorem. Let c7(X) be a Wiener algebra with approx-
imate units. If I is a closed ideal of 6(X), and f c 67(X)
belongs locally to I at all points of cosp I, then f E I. In
particular, I contains all functions in a(X) that vanish
near cosp I.
Proof. Let WO be any neighborhood of 0 in c(X). By
2.19, we can find a T E 6(x) n co(X) such that rf E f +V 00Let K be the support of T. Clearly, Tf belongs locally to I
at infinity. If a czK n(cosp I)c, then Tf belongs locally to
I at a, by 2.8. If a cKfncosp I, then f belongs locally to
I at a, by assumption; hence, iTf belongs locally to I at a.
Therefore, Tf belongs locally to I at every point of X and
at infinity. By 2.5, if c I. This shows that I n (f +V0) #for any neighborhood 40 of 0 in 6(x). Since I is closed,
therefore f E I.
2.21. Theorem. Let 6(X) be a Wiener algebra possessing
approximate units. If E is a closed subset in X, then the
smallest closed ideal JE in 67(X) with cospectrum E is the
closure of the ideal of all f c c(X) vanishing near E.
Proof. Consider I = (f c 67(X) : f vanishes near E}. Then
I is an ideal, and cosp I= E (the proof is similar to 2.10).
Also, cosp I = cosp I = E (the proof is similar to 2.14).
The theorem then follows from 2.20.
2.22. Definition. A topological standard algebra 67(X)is said to satisfy the condition of Wiener-Ditkin if for
every point acX the following holds: for any function
f c 67(X) vanishing at a, and any neighborhood'U0 of 0 in O(X),there is a T EO(X) such that (i) Tis 1 near a, and (ii) fT-E 00
25
Remark. The function T in the above definition can be
chosen even, in addition to satisfying (i) and (ii),
vanishing outside any pre-assigned neighborhood Ua of a.
For, by 2.2, there is a Ta vanishing outside Ua and equal
to 1 near a. Then, given f and ?70 in (X), there is a
T ' E 6(X) such that T' is 1 near a and (f Ta) TE Vo. Thus,we can take T = Ta T
2.23. Theorem. Let I be a closed ideal in a topological
standard algebra 6(X) satisfying the condition of Wiener-
Ditkin. Suppose f E :(X) vanishes on cosp I, and let P(f,I)
be the set of all points a in X such that f does not belong
locally to I at a. Then P(f,I) is a perfect set contained
in (Bdr cosp f) n (Bdr cosp I).
Proof. Clearly, f belongs locally to I at every in-
terior point of cosp f (and therefore at every interior
point of cosp I, since cosp f D cosp I). Also, f belongs
locally to I at every point of X-cosp I = (cosp I)c, by 2.8.
Therefore, if a E P(fI), then a/ (Int cosp f) U (cosp I)cSince
X = (Int cosp I) U (Bdr cosp I) U (Int (cosp I)c)
and
X = (Int cosp f) U (Bdr cosp f) U (Int (cosp f)c)
and notice that (cosp f)c c (cosp I)c, therefore
P(fI) c (Bdr cosp f) n (Bdr cosp I). Furthermore, if
b cX-P(f,I), then f belongs locally to I at b. Hence,
26
there is an cf E I and an open neighborhood Ub of b such that
fb coincides with f in Ub. Thus, Ub c X-P(fI). This proves
that X-P(f,I) is open and P(f,I) is closed. It remains to
show that P(f,I) has no isolated points. Suppose
a c(Bdr cosp f) n (Bdr cosp I), and there is a neighborhood
Ua of a such that f belongs locally to I at every point of
Ua - (a). Let Ta c E(X) be such that Ta is 1 near a, and has
a compact support contained in Ua. Since a c Bdr cosp f c cosp f,
f Ta vanishes at a. Given ?40, a neighborhood of 0 in a(X),there is, by the condition of Wiener-Ditkin, a T E 6(x) whichis 1 near a and (fTa) T E %. It follows that fTa (fTa)T
vanishes near a and outside supp Ta. Since f belongs locally
to I at every point of Ua - [a), so does fTa (fra)T. There-
fore f Ta - (fTa)T belongs locally to I at every point of X
and at infinity. By 2.5, fTa~ (fTa)T EI. Moreover,
fTa fTaT + ffTa fTaTE fTa -f T}a T+7V0.Since V 0 is arbitrary and I is closed, hence fT c I. Since
Ta is 1 near a, therefore, f belongs locally to I at a.
Thus a /P(fI). We have then proved that P(f,I) has no
isolated points.
2.24. Definition. A set A in X is said to be a scat-
tered set if A contains no non-empty perfect subset.
2.25. Theorem. Let a(X) be a topological standard
algebra satisfying the condition of Wiener-Ditkin and let I
27
be a closed ideal of (X). Let f be a function in c7(X)vanishing on cosp I and with compact support. If
(Bdr cosp f)fn (Bdr cosp I) is a scattered set, then f is in I.
Proof. By 2.23, P(fI) = O. Therefore, f belongs
locally to I at every point of X and at infinity. Thus, f E I,
by 2.5.
2.26. Theorem (Generalization of Wiener's approximation
theorem). Let 6(X) be a Wiener algebra with approximate
units and satisfying the condition of Wiener-Ditkin. Then
a closed ideal I of a(X) contains all functions f c i(X)
vanishing on cosp I such that (Bdr cosp f) n (Bdr cosp I) is
a scattered set.
Proof. By 2,23, P(fI) = 0. Therefore, f belongs
locally to I at every point of X, and in particular at every
point of cosp I. Hence, by 2.20, f e I.
Remark. From 2026, we have, in particular, a generaliza-
tion of Wiener's theorem for ,7(Rn) (cf. 1.12).
CHAPTER III
WIENER'S THEOREM IN L 1 (G)
Throughout this chapter, G will always denote a
locally compact Hausdorff abelian group. As a well-known
result, we know that on G there is an "essentially" unique
translation invariant Haar measure which induces a Haar
integral. The set of all complex-valued integrable func-
tions with respect to the Harr measure is denoted by L (G),
and the integral of a function f EL1(G) is denoted by
ff(x)dx, or simply ff(x)dx, if there is no confusion. TheG
L1 -norm on L1 (G) is defined by ||f|| 1 = flf(x)fdx, for eachG
f EL1(G). Similarly, the space LP(G) and LP-norm for l<p< coare defined by the usual way. For our further discussion, a
fixed Harr measure on G will be assumed.
3.1. Definition. Let T be the one-dimensional circle
group, T = (z E C:IzI = 1}.
(i) A continuous homomorphism of G into T is called a
character of G.
(ii) The set of all characters of G, denoted by 8, is
called the dual group of G. The addition in " is defined as:
for Xy^ca, +y(x) = (x).2(x) for each x c G. The identity
28
29
of G is denoted by 0. Instead of using Z(x), we shall let
Kx,_> denote the value of the character x EG at x e G.
3.2. Theorem. -, endowed with the compact-open topology,
is a locally compact Hausdorff abelian group.
Proof. (i) First, we prove that a, endowed with the
compact-open topology, is a Hausdorff abelian group. Since
the compact-open topology is finer than the usual product
topology, and since T is Hausdorff, therefore G c T with
the compact-open topology is Hausdorff. It is clear that 8
is an abelian group. It remains to show that the map
(X,9) - -y is continuous. Let us write, for each compact
K in G and each open U in T, (K,U) = {^ E:x(K) c U). Let
W be a neighborhood of x -Y. Then there is an open neighbor-
hood U = (K,U) of 0 in G, where K is compact in G, and U is
an open neighborhood of 1 in T such that x - y^ +U U cW. Find0
V cG such that V is a symmetric open neighborhood of 1, and
V - V c U. Let V x = +(K,V), and + = +(K,V). Then xcY,,y x
y C V^, andy
V^ - V^ =[ + (KV) ]-[ + (K, V)]x yy
A
= x - y + (KV) + (K, V)
C x - y+ (KU)
c W.
(ii) Next, we prove that ^G is locally compact. It
suffices if we can find a compact neighborhood of 0 E G. For
30
simplicity of the proof, we identify at this moment T as the
quotient group R/'Z = T', where R is the additive topological
group with the usual topology, and Z is the subgroup of in-
tegers. Let us denote, on T/G, k for the compact-open topology,
and p for the point-open topology (i.e., the usual product
topology). Let U be a neighborhood of 0 in G such that U is
compact. Let V = (x E T0' 0:5x <1 or <x <1J. Then V is an0' 3 "JA A
open neighborhood of 0 in T'. Let V = (u,V). Then V is an
open neighborhood of 0 in G. We want to show that V is com-
pact in G. For this, we first show that V is equicontinuous.
Let E> 0 be given. Choose m large enough so that -- < c.3m
Let W be a symmetric neighborhood of 0 in G such thatmE2W.c U, W. = Wj=l 0
for all j = 1,...,m. Hence, if xeW, then jxcU for all
j l.,...,m. Let xV, and write KxX>= y. Then jyEV for
j= l,...,m. This implies that 05y <y, or 1 - <y1.
Therefore, for all xceW and x 1,K|x,2)>-K(,x^)|<i.-< c33m
This proves that V is equicontinuous at 0 E G, and hence atevery point of G. Therefore Vk p c G is also equicon-
tinuous. Since T ' is compact, therefore for each x cG, the
closure of the set T[xi = {(x,> : xcV) is compact in T'.
Hence, by Ascoli's theorem (see [1]), V is k-compact.
3.3. Theorem. Suppose 1l5p<co., and fcELP(G). The map
x-> Lxf is a uniformly continuous map of G into L (a).
Proof. The proof is similar to 1.5.
3.4. Definition. Let fg be two measurable functions
on G. We define their convolution f*g by the formula
(f*g)(x) = If(x-y)g(y)dyG
provided that fJff(x-y)g (y)J|dy < .
We state, without proof, the following basic theorems.
For the proofs, refer
3.5. Theorem. (f*g(x) = g*f(x).
(ii) If fcEL (G)
uniformly continuous.
(iii) If f and g
A and B, respectively,
so that f*g eC00 0(G)
(iv) If 1< p < C,
then f*g CO(G)
to Rudin's book [21.
i) If f*g(x) is defined, then
and g cL*(G), then f*g is bounded and
are in a00 (G) with compact supports
then the support of f*g lies in A +B
1 1,np + q 1, fcELP(G) andgEL()
(v) If f and g are in L (G), then f*g E L (G), and
l f*gll : 11f 11, * 1.
(vi) (f*g)*h = f*(g*h) for fg,h E L1 (G).
The well-known fact that L (G) is a Banach space com-
bining with 3.5 (i), (v), and (iv), and some other simple
properties give the following theorem.
3.6. Theorem. L (G) is a commutative Banach algebra
with convolution as multiplication.
3~2
3.7. Definition. The Fourier transform of a function
f Ll (G) is defined as the complex-valued function ? on the
dual group 8 such that
= ff(x)K<-xi >dx.G
3.8. Theorem. Let f,gcL (G).
(i) 1 |ff||1.
(ii) (af+Pg) =a + $9, for a, P E C.
(iii) (f*g) = . ^.
(iv) (Laff"(2) = K-a, Xf(X), for aEG.
(v) If a E G, then ( p f)^ = LJf, where cp,(x) = Kx,^>.
(vi) f = f, where f(x) = f(-x).
Proof. The proofs are simple, and we shall prove (iii)
only.
(f *g) (X)= f(f*g)(x) K-x, >dxG
f Jf(x-y)g (y) dyK<-x,2> dxGG
= fg (y)-yb2>dy fK-x+y,X> f (x-y) dxG G
=2(^)?(x^).
We shall state, without proof, the following important
theorems which can be referred to in Rudin's book [21.
3.9. Theorem (Riemann-Lebesgue lemma). The Fourier
transform of a function feL1 (G) vanishes at infinity. That
Ais, given E > 0, there exists a compact set K c G such that
f(x)l < E for all x c K^R
133
3.10. Theorem (Uniqueness theorem). A function fcEL1 (G)
is uniquely determined a.e. by its Fourier transform. That
is, if fcL (G) and f(^) = 0 for all X^ 0, then f(x) = 0 a.e.
3.11. Theorem (Inversion theorem). There is a normali-
zation of the Haar measure on 0, say d2, such that for all
functions fcEL1 (G) which have a Fourier transform PEL (G)
the following relation holds.
f (x) = 5fx,x> P(2)dx, for almost all x c G.
3.12. Theorem (Plancherel's theorem). There is a
normalization, the same as the one in 3.11, of the Haar mea-
sure on 6 and a bijective linear map f-. f of LI(G) onto
L 2() with the following properties.
(i 1|f||12 1? 2'
(ii) If fcL (G) n L (G), then f^(2) = ff(x)K-x,2)dx, a.e.1l 2 Gon G. If fcLQ() n L (G), then f (x) = Sx, ̂ >( )dA, a.e. on
G .
Remark 1. The function ? which we get from the
Plancherel theorem for-a given f L2 (G) is usually called
the "Plancherel transform," which coincides with the usual
notion of the Fourier transform for fEL1 (G) n lL2 (G). The
symbol ? will be used for a Fourier transform as well as a
Plancherel transform whenever it makes sense.
Remark 2. The set of all Fourier transform of func-
tions in L1 (G) is denoted by 71(1). We define a norm on
91L(a) by Jj1 = 11f11 1, f E:L1 (G). Then, 91(G) is a commutative
Banach algebra with the usual algebraic operations. More-
over, the map f - 9 from L (G) onto 71(t) is an isometric
isomorphism.
13.13. Theorem. If f ,f2 EL (G), then
(I) ff 2(x) dx =f 2 ()^d5
and
(ii) [If-f 2 f1*2'
Proof. (i) follows easily from 3.12 (i), and the rela-
tion between the L -norm and the inner product in L2-space.
(ii) We prove first that (ii) is true for f2 EL (G) L2 (G).Replacing f2 by 72 in (i), we get
Jf(x) f2 (x) dx = Sf 2 (-x) d5.
Again replacing f2 by (Cpa)f2 in the above, where & G_, and
cp (x) = K-x,a>, we obtain-a
f1(X) f2(x) K-x,a> dx = 5f()2(9-x) d^.G .
A ~1 2Therefore,[f-f2( ) f1* 2(a) for f2 cL (G) AL (G).
Now suppose, in the general case, f ,f2 c L (G). Choose a
sequence {g n} cL (G) n L2(G) such that IIg n~f 2 2- 0 as n -co.
Then
1f-ff 4r- 11 If f *I1ACf2 1f'2 2,r- f1* n1 Io+ f~*^gn~ f2
if f2 - n1 o + 1 * (gn -~f2) 1 c
:51f~ 2 -n) ll1 12 ~n f2 12
<If 112 1f2-gn11 2 + 1 n112 12gn-f 212 - 0
as n -+ c. Therefore, [fa. f 21 f 1*if 2
35
Remark 3. Notice that the "^" on the lefthand side of
(ii) denotes the Fourier transform, and the "a" on the right-
hand side denotes the Plancheral transform.
3.14. Theorem. 1( ) consists precisely of the convolu-
tion F1 *F2 , with F1 ,F2 EL (G).
Proof. Suppose F1 ,F2 L(E): . Write F1 = ?F, F2 f2.with f ff2 EL (G). Then by 3.13 (ii), F1 *F2 1* 2
[fw f 2 r C f2 ^ E 1(a)
On the other hand, if f EL (G), then we can write
f = ff 2 with f1 ,f2 E L2 (G). Therefore, f = f f2 1*f2
where f, EL2
2.15. Theorem. Let ^, EL2( ) with supports containedA 2in a compact set K. Let g,hcL (G) be their inverse
Plancherel transform, respectively. If f = g-h, then
(i) f = gA
(ii) IlLff-| 2ax |K<y,> 1XER
Proof. The proof of (i) is simply from 3.13 (ii).
(ii) Write Lyf-f = (Lyg-g)Lyh +g(Lyh-h). By Holder's
inequality, and Plancherel's theorem,
A(L g-g)L h|L | |Lyh|2
(Lyg-g) ||2 |||| 2 = ll(Lyg) ^g||2 I1hJ1 21
= 2 111 1
$() |-y,x>-|2dx^)||||2
:51gJ1 2 1lhll2 max I(y<Y > -XEK
Similarly,
g (L h-h)||11 5 &2 2max | (y,^) 1>XEK
3.16. Theorem. Let , be a compact symmetric neighbor-
hood of O in ^. Let 9 be the characteristic function of 2,
and f, be the inverse Plancherel transform of 0.Let
o(x) =an(x) m(1)[fjx) ]2 , for each xcEG, where m denotes
the normalized Haar measure on 9. Then,
(i) a(x) 0 for all x c G and fa(x)dx = 1,
(ii) |JLa - a 5 2 maxJ| (y,x^>- 1,for y EG.XeS
Proof. (i) By the inversion theoremf(x) =
f x,X> (2)d = f(x,x^>d. Since 8 =
f (-x) = ,fK-x,> O(x)d
K-x,> dx = S x,>d
S A= .x>)dx.
Therefore, f,(-x) = f (x) and f(x) is real. By definition,
G(X) 1 2f(x)2 0 for all x EG. By Plancherel's
theorem, |f9 2 = M(S). Hence,
(X)12Ja(x)dx = 1 2dx = I 2 - I.
m(f) fxm('(ii) Applying 3.15 (ii) with g = T , = Og,K S, we get
L a-c|I : 2j# 1 max | y,9>-1|y 12 m(g) A
= 2 max |xy,5^) -lj.
37
3.17. Theorem. Let 9 be the Fourier transform of
the function a,- obtained in 3.16. Then 0 : 5 1, <() = 1,S S
and supp ^a c 9+.
Proof.
m(X) (0*)()
0() S
dy =m((X^+^)4%
( ) (x+S)n$ m(y
Hence, 0 :5 < 1, and 9Q(6) = 1. Moreover, if o (x) / 0,A Athen (X^+S)n^ / i. Therefore x E S+S, since S is symmetric.
It follows that supp ^ c S+2., since 9 is compact.A12
.Remark. Since G is locally compact, we can take S
arbitrarily small. The functions ag and their translates
show that the algebra 71(G) has the second property of a
standard algebra.
3.18. Theorem. Let 9 and T be two compact symmetric
neighborhoods of 0 in G. Let ,$^ be the characteristicS +T
functions of $ and S+T, respectively. Let fV, f be the
inverse Plancherel transform of - and 0 ^. Define9 9+T
T(X) = T A(x) 1f (x) f ,(x), for each x EG. Then,T m(S) S+T
(i) TEL1(G) and IT1l1m:5) 2
(S)lm(^)
(ii) JJL y Tlll 2 fM ) 2 ^max |(y,x> -11, for ycEG.m(') xE+T
Proof. (i) Since f and f,.^ are in L2 (G), therefore
T cL1(G). Moreover,
11 T 11 11f 12 1 ^fA ̂ 12 12 1-0m(^+T)22m()S S+T m(2) +T m
(ii) Using 3.15, with g = h hK = S + T, we
have
JL T-T|| 5 2 - s12 210max y^> ,m(S) 1$+T
1
= 22 }m xx2 ^1(+T 2malyA>l1m(s) xcS+T
3.19. Theorem. If is the Fourier transform of
the function T obtained in 3.18, then 0 : <1,
T ^() = 1 for all x^c T, and supp c +S+ T.
Proof. The proof is similar to that of 3.17.
Y 4(2) (# * Om( x S nS+T)T( )+T m(()
Hence, 0 T < <1, Y (^)= 1 for all xc ET, andST STsupp T AA c S+ S+ T.
S,TRemark. Theorems 3.16 through 3.19 may be viewed as
the generalized result of 1.3 in Chapter I.
3.20. Theorem. If K c G is compact, then for every
C> 0, there is a non-negative function aE L (G) such that
Sa(x) dx = 1, and iLy a -a |<c for all ycE:K.
Proof. Let 2 be any compact symmetric neighborhood of
0 contained in the open neighborhood (K,U), where
U = tz c T:fz-l|< L). Let a = a.. Then the results follow2S
from 1.16.
39
3.21. Theorem. If a >1, then given any compact set
K C G , there is, for every c > 0, a function T E L (G) suchthat
(i)11 T jl < a, and T is 1 near 0 in
(ii) IL T-T,| for ally c K.Proof. Given aK, and c> 0, let
U = (xEG: yX>-1< 1 (/:/a) for all yEK}.
Let S be a compact symmetric neighborhood of 0 contained in
U. Then we can choose a second such neighborhood ^' satisfying
S+T cU and m(S+T)< am(s^), since U is open and a>l. Define
T = TA as in 3.18. ThenST , 1
L T-T11 5 2 max | )XES+T
21 E< 2(a) g(-. <E.
Furthermore, (X) = 1 for all 2cT, by 1.19.
3.22. Theorem. Let o be a subset of L1 (G) having thefollowing properties.
(i) d is bounded in L (G); that is, there is an M> 0such that ||al|II5 M for all a E d.
(ii) If K is a compact set in G, then for every c> 0,
there is a function a E P such that |ILy a -all < c for all y K.
Then, given any fcEL (G), there exists, for every E> 0, a a c dsuch that |ff* a - [ff(x)dx]a ill < E.