I ntnat. J. M. Math. Sci.
Vol. 5 No. 3 (19821 441-457
441
SECOND DEGREE POLYNOMIALS AND THE
FUNDAMENTAL THEOREMS OF HARMONIC ANALYSIS
KELLY McKENNON
Department of Pure and Applied Mathematics, Washlngtdn State University
Pullman, Washington 99164
(Received September Ii, 1981)
ABSTRACT.
The concept of a second degree polynomial with nonzero subdegree is in-
vestigated for Abelian groups, and it is shown how such polynomials can be exploited
to produce elementary proofs for the Uniqueness Theorem and the Fourier Inversion
Theorem in abstract harmonic analysis.
KEY WORDS AND PHRASES.
Fourier inversion theorem, second degree polynomials.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE.
INTRODUCTION.
i
Let G and M be fixed, but arbitrary, Abelian topo-
Second degree polynomials.
logical groups.
Let HOM(G,M) be the group of continuous homomorphisms of G into M,
Let AFF(G,M) be the set of all functions
equipped with the compact-open topology.
G
43A45, 12A20.
b+f(x) such that b M and f E HOM(G,M).
x
tinuous symmetric maps
is in HOM(G,M).
IGxG
f(x+y)
f(y)
Second d_er_ p_91_i_al_s
f(x+y+z)
for all x, y, z
G
with nonzero
i.e. continuous functions
f(x)
0
f(x+z)
we write
fig
for all x, y
wit___h nonzero, s..u.b.degr.ee,
f(x+y)
set of con-
M such that, for each a E G, the function G
Firs__.t degre e polynomials
elements of HOM(G,M):
SBH(G,M) for the
Write
P2(G,M)
x+ (x,a)
subdegree are, by definition,
M such that
G.
(I)
are defined to be continuous
f(y+z) + f(x) + f(y) + f(z)
0
(2)
for the set of all such.
The following observations are elementary:
P2(G,M)
is closed under (pointwise) addition;
(3)
442
K. MCKENNON
f(0)
For
0 for all f
EP2(G,M);
HOM(G,M) c P2(G,M).
fiG + M, define f*IG M by f*(x)
(4)
(5)
f(-x)
/
for all xG.
Clearly
f*
A function
functions
EP2(G’M)
for all f
fiG M is called even
in P2(G). Inserting x
P2(G,M)
if f=
For
If
P2(G’M)’ (7) implies
f (EP2(G)
f(2x)
f (P2(G,M) is even and x=-z
Write
EP2(G)
for the set of all even
-z in (2), we obtain with use of (4)
y
3f(x)-f(2x) + f(-x)
f*.
(6)
O.
(7)
f
2f(x) + 2f(y)
4f(x)
for all xEG.
is put into
(8)
(2), then
f(x+y) + f(x-y).
Conversely, if (9) holds for f
P2(G,M)
inserting
(9)
xffiy
in
(9) yields
f(2x)
4f(x).
Thus (9) is equivalent to the evenness.
If
fiG
M satisfies
of all odd functions in
f
f*=-f,
P2(G,M).
60P2(G,M)
f(2x)
Putting x =2t+y and z =-2t-y
2f(y)
it is said to be odd.
For
(P2(G,M)
f
2f(x)
Write
OP2(G,M)
for the set
(7) implies
for all x EG.
(I0)
in (2), one obtains
+ f(2t+y) + f(-2t-y) -f(2t+2y) -f(-2t)
0.
(ii)
If f is odd, (II) reduces to
2[f(y) + f(t)
f(y+t)] =0 fOr eli t, yG.
(12)
Thus we have
=EOP 2(G,M)
2 f HOM(G,M).
(13)
In view of (I0), (12), and (5), (13) can be improved in special cases:
f
(0P2(G,M)
f EHOM(G,M)
(Vx (G) (3y(G)
if either
2y=x
(14)
(14i)
or
M has no element of order 2.
(141i)
In particular (14) holds when G is compact and connected ([2] 24.25) or when M is
torslon-free.
443
FUNDAMENTAL THEOREMS OF HAP4ONIC ANALYSIS
It is not true that (14) holds in general.
Let Z be the additive group of
integers and C 2 the cyclic group of order two with generator I.
plZ
Z
C
2
by
p(m.n)
That p is odd is clear.
If
x=mn,
Define
if
y=
f
if both m and n are odd
0
otherwise
P2(Z
That p is in
9
Z,
C2)
(15)
is evident from the following.
z =MN, then
and
m,n even
others arbitrary p(y+z)
0
m even
0
0
0
p(y)
p(z)
1
0
0
0
0
1
0
0
0
0
1
0
0
1
0
0
0
0
0
1
0
m,
even
others odd
1
0
0
1
0
0
0
m,M even
1
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
I
1
0
0
0
0
1
0
0
0
0
0
0
1
1
0
0
0
0
1
0
0
0
0
0
0
0
0
others odd
m, 9 even
others odd
m,N even
others odd
m,,N even
others odd
others odd
m, ].t, ) even
others odd
m,,N even
others odd
m,,N, even
others odd
m,M, even
others odd
p(y)
..p(z)
p (y+z)
1
m,M,N even
others odd
m,M,v,N
even
others odd
m,,M even
others arbitrary
The cases in which n is even can be found from the above table invoking the symmetry
of P.
Thus
pEOP 2(zz, C 2)
If
is in
SBH(G,M)
and
and
f(x)
p
HOM(ZmZ,
(x,x)
(16)
C2).
for all x E G, then f is said to be
quadratic-----the set of all such will be written QD(G,M).
That
444
K. MCKENNON
(17)
QD(G,M) c EP 2 (G,M)
On the other hand, consider the function p defined in (15) and
is elementary.
p(x)
assume that
(x,x)
x E Z
for all
=p(leO)+ 2(1,0,01) + p(Ol)
O,
0+0+0
Thus
an absurdity.
p E EP
6
Then
+ (1@0,0@1) + (0@1,1.0) + (01,0@1)
1= (p(ll) =(1@0,1’0)
For
Z.
2(ZgZ, G2)
SBH(G,M)
but
(x,x), the equation
f(x)
and
(18)
QD(G,M).
p
f(x+y)
f(x) + f(y) + 2(x,y)
shows that
(Vx, y (G)
f (HOM(G,M)
2(x,y)
There is a partial converse to (17).
h#1GxG
(x,y)
+
as follows readily from (2).
(8) yields
f(x)
h
#(x,x)
If h is even and
x
G.
{2h h E EP (G,M)}
2
Any function f
(h + h*) + (h
f
(P2(G’M)
h*)
f
then
(20)
SBH(G,M)
2h, a direct computation with
Thus
c
QD(G,M)
2h
of the form
for h
P2(G’M)
can be written
(21)
+0P2(G)
TRANSLATION-PSEUDO-INVARIANCE.
For
fiG- M
G.
for all x
function
and a(G
fig
THEOREM I.
(x,y)
f,a
is defined by f (x)
f(x-a)
A
AFF(G,M) such that
ef’a + f.
a
(22)
The TPl-functlons f such that f(0)=0
Let f
P2(G,M)
be arbitrary.
Define
f(x+y)-f(x)-f(y) for all x,yG.
is in
IG- M
M will be said to be translatlon-pseudo-invarlant (TPI) if, for each
f
PROOF.
the translation f
Thus the only translatlon-invariant functions are constants.
a E G, there exists
that
is in
P2 (G,M),
and so
2h EP 2(q)
2.
If h is any element of
h(x)-h(y)
h(x+y)
for all
(19)
0.
SBH(G,M).
f (x)
a
P2(G,M).
IG x G + M by
Direct computation with
Thus, for all a,x (G,
f(x-a)
are the elements of
(x,-a) + f(x) + f(-a)
(2) shows
FUNDAMENTAL THEOREMS OF HARMONIC ANALYSIS
445
so
f
uf,a+
a
f where
uf’alG
f(-a) + (x,-a).
9 x
Now let f be an arbitrary TPl-function.
g(a)
M and
a HOM(G,M)
a
For a,b,x
G, there exists
Ha(x) +
f(x)
for all
x G.
G we have
a+b
g(a+b) +
g(a) +
Then, for each a
such that
g(a) +
f (x)
(23)
a(-b+x)+
(x) + f(x)
g(a) +
f(-b+x)
fa(-b+x)
fa+b(X)
a(-b) + Ha(x) +
g(b)
+b(x)
+ f(x),
whence follows that
g(a+b) + N
a+b
g(a) + g(b) +
Ha(b) + n a + nb
(24)
Since the left side of (24) is symmetric in a and b, the right side is as well
Thus
a
Consequently the function
SBH(G,M)
Hence
a+b
g(a+b)
For x,y,z
b (a)
(b)
a
for all a,b
IGxG
+
b
X(y)
(x,y)
g(a) + g(b) + (a,b)
(x+y,z)
P2(G’M)"
G.
for all a,b
g(y+z) + g(x) + g(y) + g(z)
(x,z)
0
Furthermore, for each x
G,
=f_x(0)
[g(y+z)-g(y)-g(z)]
[g(x+z)-g(x)-g(z)]
(y,z)
f(x)
3.
fact, in
G, the above yields
[g(x+y+z)-g(x+y)-g(z)]
g 6
is symmetric and is, in
so (24) becomes
g(x+y+z)- g(x+y)-g(x+z)
so
G.
=g(-x) +
-x(0)
+ f(0)
gx).
Q.E.D.
THE FUNDAMENTAL THEOREMS OF ABELIAN ABSTRACT HARMONIC ANALYSIS.
In the sequel G will be a fixed, but arbitrary, locally compact, Abelian,
Hausdorff, topological group (LCA gp.)
Haar measure on G,
LP(G)
for the
For any such group G we write
LP-space
relative to
%G
(p
[I’] )’
%G
and
for the Banach space of bounded, complex, regular Borel measures on G and
for the set of continuous, positive-definite functions on G.
complex numbers of unit modulus and
HOM(G, T).
for a
(G)
(G)
Let T be the group of
any LCA group topologically isomorphic to
K. MCKENNON
446
The fundamental theorems of Abelian abstract harmonic analysis are as follows:
(DUALITy THEOREM)
T such that, if for a G
to
(25)
There exists a continuous bihomomorphism e from
I
y
a
Y
the a and b are defined by
b
and
and
then the maps
G )a
HOM(, T)
b
and
b
HOM(G,T)
are both topological isomorphisms;
(G),
(UNIQUENESS THEOREM) if, for each
f-x’y d(y)
Y
is nonzero whenever
then
(26)
is defined by
G
is nonzero;
(INVERSION THEOREM) if, for each f
L
I(G)
and h
LI(),
and
are
(27)
defined by
I
%
then
f f(t)-
Y
tey
G
G
and
x
can be normalized such that, whenever
(PLANCHEREL’S THEOREM)
f L
d%G(y
2(G)
L
%
f h(s)
is in
x e s
LI()
d%
for f
(s),
LI(G),
can be normalized such that, for each
then
(28)
I(G)
G
lfl 2
(BOCHNER’S THEOREM)
d%
ll
G
G)
{
2
d%
() +)
(29)
There are two basic approaches to the proof of the fundamental theorems.
The
more modern approach is based on the Gelfand-Naimark theory of commutative Banach
It is based on the observation that, under the convolution
Algebras.
operation
f
,
,
defined by
h(x)
I
f(x- t) h (t)dt
for all f, h 6L
I(G)
and x 6 G,
(30)
G
LI(G)
is a commutative Banach algebra whose spectrum is topologically isomorphic to
}{OM(G,T).
This approach was taken by D.A. Raikov in
[5].
Here Bochner’s Theorem
(29) is established first, and the other theorems deduced from it (see [I] and [6]
for more recent accounts).
The original approach was based on the structure theory of LCA groups as
developed by L.S. Pontryagin and E.R. van Kampen.
Theorem (25) in full generality
FUNDAMENTAL THEOREMS OF HARMONIC ANALYSIS
"wa.s
first published in 1935 by van Kampen
[7].
447
The other fundamental theorems are
then deduced from (25) by judicious use of the Stone-Weierstrass Theorem and some-
what delicate construction of approximate identities for
theory (see
[2]
[3]
and
LI(G)
based on the structure
for instance).
Our approach here is based on the structure theory and takes (25) for granted.
It however is not dependent on the Stone-Weierstrass Theorem and the approximate
identity exploited is relatively simple.
functions and simple complex analysis.
The basic techniques are use of TPIThe theorems (27), (28) and (29) are
essentially equivalent in the sense that, once one of them is established, the task
of establishing the others is relatively simple.
Consequently we shall employ our
method to prove only (26) and (27).
Let
T
be the multipllcative group of nonzero complex numbers.
We shall make
exception to our previously adopted convention by employing multipllcative (rather
than additive) notation for T
G
of G to be an abstract group isomorphic to HOM(
G
and extend o to a map on
a
and its subgroup T.
I
y
We define the complexificatlon
,T
such that, if for each
)
a
containing G as a subgrou
G
a is defined by
a e y, then
G
In the special case
)
HOM(G,T
a
G= T, we have
zo n
z
n
Z and
for all
z
T
and n
E and E
(32)
Z.
As a second example, consider finite dimensional real Hilbert space E.
G
(31)
is an isomorphism.
Then
can be taken any complex Hilbert space in which E is a real
form--here
e _8
e
-i
<,B >
for all
E
8
A function f which is the restriction to G of a function
fo
is analytic for each 6
(33)
E
fl G
such that
AFF(C,G),
(34)
is said to be entlre--the Uniqueness Theorem for analytic functions implies that
the function
f
is unique
For f entire and a
f by
f
a
G 9x
f (x-a)
G
we define the translation f
a
of
448
K. MCKENNON
,
The complexification
C,
the concept of an entire function defined on
and
its translations are defined analogously.
LI(G)
We write
AFF(,G
each
(where
)
is the
LEMMA I.
Let
for the set of all entire functions f on G such that, for
> 0,
and
sup{llfo(z)lll: Izl _< 6} <
Ll-norm). In particular L I (G) c LI(G).
f L I(G) and a G
be arbitrary. Then
f f a d%G
G
PROOF.
ll
HOM(,]R+)
each
z
,
o(z)
ll
the element
o(z)
Let
c G
such that
c
a
c
and define T{ AFF(, G
c+o(z)
T(z)
for all
z
Then (35) implies that, for any simple closed contour y in
I
YG
If (x-
for
expCz in
Y
be such that
o
Define
G
of
ll" __a where
HOM(,G ) by letting,
y can be written
a
Y
HOM(,T).
and
be
(36)
f f d% G
G
l
The function
(35)
by
,
d%GCx) dz <
(z))
so it follows from Fubini’s Theorem and Cauchy’s Theorem that
f f f(x-T(z))
yG
Morera’s
analytic.
d%G(X) dz
d%G(X)
f f f(x-T(z)) dz
GY
Theorem now implies that the function
FI z
-
For imaginary z, (z) is in HOM(,T) so (z)
f f(x-T
G
f0d%G=0.
G
(z))d%G(X)
is in G--hence
is
T(z)
is
in G as well and we have
F(z)
f
G
f fd%G
fT(z) dG
by the invariance of the Haar integral.
The Uniqueness Theorem for analytic
functions now implies that (37) holds for all
But
T(0) =a, so (36) holds.
(37)
z (
In particular
f f dA G
G
Q.E.D.
Relative to study of the Fourier Transformation, the utility of
dependent on the size of
F(0)
P2(G,T) LI(G).
P2(G,T)
This size can vary, as is shown by
is
449
FUNDAMENTAL THEOREMS OF HARMONIC ANALYSIS
Theorems 2 and 3 below.
LEMMA 2.
f
Define
E
Let E be finite dimensional real Hilbert space and k
]0,I[.
C by
k<x’x>
f(x)
Then f is in EP
2(E,T)
L
I(E)
for all
x E.
and
x
< x
l--k’ l--k >/4
k
f (x)
(38)
for all x( E
(39)
(where E is the dual of itself as in (33)).
PROOF.
That f is in
Let E
E
and e
follows from (17).
be as in (33) and let
leaving the elements of E fixed.
Lemma
analytic so f is entire.
llflll
EP2(E
f
k<t
-iy,(t iy)>
E
Then h(K)
fIE
z
k<Z z>
is
f k <t’t>e-21<In k y,t> k-<y,y>
dE(t)
E
dE(t)
Q.E.D.
Let K be a compact subgroup of G and h an element of
P2(G,TC).
T.
PROOF.
there
From (13) we have
group of T
p#1G
h2(K)cT (where h21G x- h(x)2).
exist p EP2(G T) and g
0P (G,T) with h 2 --p" g.
2
g2 HOM(G,T), whence follows that g(K) 2 is a compact sub-
It will suffice to show that
In view of (21)
Let
Then the function
implies that, for each y (E,
Letting x--21nk "y, we have (39).
LEMMA 3.
be the conjugation of the HJlbert space
C
Thus g(K)
xG
T
T
be defined by
#
p (x,y)
From (20) we have that
subgroup of T
p(x + y)
p (x)p (y)
p#( SBIt(G TI)
and so a subgroup of T.
#
p (x,x)
p(2x)
p(x) p(x)
for x,y
G
Thus, for each x (G,
Furthermore, by (8),
p(x)
2
whence follows that p(K) c T.
Finally,
h
2(k)
c p(K)
g(K) c T" TfT.
Q.E.D.
p#(x,K)
is a compact
K. MCKENNON
450
There are LCA groups G such that L
THEOREM 2.
lira
is void
Let G be an LCA group with a sequence Sn of compact subgroups such
PROOF.
that
I(G) P2 (G T)
(such as a discrete group which contradicts Burnside’s
)tG(Sn)=’
P2(G,T), Lemma 3 implies that h(Sn) c T
Then, for h
conjecture).
for each n.
Thus
llhll I
for each
%G(G)
f
Then
(b)-- a =(-a)
f.
b for b
implies
(-a) f
G
d%G= fG ()a d%G
.
f
G
Since the
dAN
G, it follows that
a
f bd%G
G
Consequently
Similarly
Q.E.D.
Evidently we need only consider the case f
PROOF.
fact that
limn fSn lhl dG=limn G(Sn)--"
Suppose that G is compact and that f is a linear combination of
LEMMA 4.
characters.
_>
is
v must be
0
if b#O
(G)
if b
.
0
times the characteristic function of the singleton b.
%G(G)
%G(G)
Q.E.D.
be the directed set of open neighborhoods of 0 in G.
Let
THEOREM 3.
x
(A,m)
flG\S =0
f(0) =I
-
f(A,m)
and
fls
P2(S,T)
is in the convex hull of
for all
there exists B
lira
I(G)
and a surjection
satisfying the following properties.
f >_ 0
and 1
For each A
c L
There is an open subgroup S of G, a set
for all f
(40)
(41)
f
such that BoA
(42)
and
f(A,m) dXG/ f f(A,m)d% G
G\B
f
m-=o B
there exists a net f
in
such that the net
a/l]fa[[
I...-
converges
(43)
uniformly to I on compacta;
L
I()
E
f (0)
PROOF.
for each
(44)
f
h (0) for all f,h
(45)
(i.
The Pontryagin-van Kampen Structure Theorem
([2] 24.8) implies
that G
may be regarded as the direct sum of n-dimensional real Hilbert space E and a sub-
group H containing a compact, open subgroup M.
We denote EM by S and its dual
FUNDAMENTAL THEOREMS OF HARMONIC ANALYSIS
group by g.
([2] 24.11, 24.10)
The Duality Theory
determined by the equation
IS
o(y)
for all
{y
is a topological epimorphism, that
Z-- {y { :
the dual of E, and that
The map
o[
+
epimorphism of G
y
IE
{
451
implies that the map o
S
,
(46)
"
y
lM
i}
may be identified with
i} may be identified with the dual of M
has a canonical extension (also denoted by ) to an
S
such that (46) holds for all y {
g
onto
C.
Since
M is compact
so
and continuous homomorphisms preserve compact groups, it follows that
y
we can (and shall) identify
as a complex Hilbert space in which E is a real form.
As in (33) we may view E
be the conjugation of E
Let
with
C
leaving the elements of E fixed.
topologically isomorphic to E, there exists an isomorphism
is
Since
81E
sending E
satisfying
to
e(y)
x
-i<x,y>
,E)
HOM(
Homomorphisms
e
e
o(y)
for all
6
and
%()
(M)
IE
that
is
that
and
are determined by the equation
(48)
for all y 6
%S
is the restriction to S of,
g(x+y)d(y)d%E(X)
for all
/ go
/ gd%
For each
direct sum S
let A
I
A
E
d%
and A
M
compactness of
M\AM
{xM: x
-
for each
that
gL(S),
(49)
y
for some
such that
(51)
be the canonical projections (relative to the
such that
there is a finite subset
1
(50)
gLl(g).
of A into E and M respectively.
EM)
there corresponds at least one y
c
%G’
%E-%-measure preserving, that %gsatisfies
for all gLl(g)
g d%g= g(x+y) d%(y)d%(x)
Weil’s Theorem ([3] 28.54) permits us to normalize
MIAM
(47)
y{ E
%E satisfies
/ /
EM
gd% S
S
HOM(,Z)
OD(y) (Y)
We may (and shall) assume that
xE
y F(A)}.
x(y)
Since to each
MAM
y(x)# I, it follows from the
F(A) of g such that
Define
x
452
K. MCKENNON
hA =Then h
+ n
1
y
yEF(A)
2
+ (_y)
2
satisfies
A
i, 1 >_ hA >_ O, and
hA(O)
For each m]N, define
f(A’M)
IGS
f(A’m)
hA(X)
< 1
for all x E
G /JR+ by
f(A’m)(x+y) =e-m<x’x> hA(Y)
0 and
f(A,m)
P2(S,T),
be the set of all such functions
Let
(53)
for xE, yM.
That (40) holds is evident from Lemma 2, that HOM(S,T) c
construction of
MAMand the
f(A,m)
That (41)
holds is obvious.
For each r > 0 let E(r) be the set {x
the change of variables
mx
s
f
e
-m<x,x>
B
E(r’) + D
f
B
lira
m<= /
G\B
m-o
m
m-n/2
e
-<s s>
I
e
Since
(54)
dz (s)"
a positive number such that
Then (54) yields
dE(S)
E(s)
-<s,s> dA
-
which proves (42).
A
-<s s>
E\E(r’
EkE(mr’)
For
1/2} and r’
e
x>
1/2 f e -m<x, dE(X)
E(r’)
>_ lira
-m<x, x>
m-o
e
I
dE(X)
f(A,m) dG
f(A,m)d G
2E / r)’
Then, for each m6,
yields
{x6M:hA(X)>
is a subset of A.
m-nl
=lim 1/2
E: <x,x> _< r}.
f
dz(X) m-n/2 E(mr)
E(r)
Let D be an open subset of
{
and r>0, let A
r
:
{y(
x
y- I < r
ttOM(,,T) bears the compact-open topology,
Let
family of compact subsets of
6 > 0, apply (42) to obtain B 9 and f
B
fdl
G
>_
A{
{
{A*:A (l,r >0}
and r {
]0,I[
x A}
is a fundamental
be arbitrary
G such that BoA and
l-___/_r f fdA
G
r
C\B
for all
For
453
FUNDAMENTAL THEOREMS OF HARMONIC ANALYSIS
so that
/ f dE
G\B
Thus, for each y A
f
G
r
f f" (l-y) d% G
f f (l-y)d% G
B
(Y) -i
2
llflll
G\B
+
llfll I
G\B
Ifl d% G
_< 3r
llfll
This proves 43.
f(A,m)
Let f=
,
for each y
be an arbitrary element of
(Y)
(--) dE G=
/ f
G
/f
.
From (40) and (46) follows that
(-y) d%
S
S
so that by (49) and (47)
f e -m<x’x>
f(y)
EM
hA(S)
e
-i<x,q (y)>
s e(-y)
(y)
Since
4m
e
<q(y) N(y)>
.c. m
,
g
)
y
e
-4 <n(y),n(y)
An element b of
[[
(h
A!M)^((y))
for all y
the function
is analytic--as an extension of
satisfies v=
-n/2
>
,
(55)
f e-<x’ x> d% E (x),
E
From (39), (54) and (55) follows, upon setting c
_I
d(s) dE(X
.
(56)
.
(4m)-n/2-(hAIM)^((y))
it will be denoted
can be written uniquely as
For such b =u+v and y
,
u+v where
u
and v
>
e
Since the support of h
A
the n=ber w(A)
is a subset of a finite set
[(hA[M)l d%
(57), and (54) yields
(57)
e
e
is finite.
us,
F(A) (-F(A))+ F(A)U (-F(A)),
application of (51), (50),
K. MCKENNON
454
-n/2
I
-Am <
f
e
f
e
(y-u) ,q (y-u) >
dl(y)
_i
4 <x x>
E
e
m<(v)
dkE(X
m< q (v)
a(v)>
I
(4m)-n/2.w(A)= e-m< (v)
"c
(58)
(4m) -n/2 w(A)
c
(v)>
e__
w(A)
(v)>
c
2
w(A)
This proves (44).
From Lemma 4 follows that
AIM )^ dX= (hAIM)^v(0)
f (h
I.
Consequently (56), (49), and (54) yield
v(0)
=i
dX
/ e
E
m<X’X> dXE(X).C.(4myn/2
(4m)n/2"<4m)-n-2"
which proves 45.
LEMMA 5.
c
2
Let S be an open subgroup of G and f an element of
^
and
1
Q.E.D.
conditions (40) and (44) of Theorem 3.
PROOF.
2
c
(hAIM)^ d%
^(0)
oI
f.
(59)
fls 6P2(S,T).
the canonical epimorphism as in (46).
exists a function
wlS
f
Let g be the dual of S
By (23) of Theorem 1 there
such that
--X
f(x) "w(x)
f
for all x6 S
(60)
For each x6 G, (51), and (60), and (36) yield
^(x)
/ S f(t)
tez
dlG(t
G f(t)
-x z
f(x) f
I f(t)
f(t)t(R)z
d%s(t dl(z)
f(x)
^
diG(t)
t.y
dlg(z)
as
f(x)
satisfying
Then
We evidently may assume that
g
LI(G)
/
S
te
f_x(t)
tez
(z+w(x))
f(x)
(0).
(-x) ,y
G f(t)
/
Q.E.D.
dl
(y)
dls(t dlg(z)
dlg(z
t,z
dlG(t dl(z)
455
FUNDAMENTAL THEOREMS OF HAILMONIC ANALYSIS
(Fourier Inversion and Uniqueness Theorems).
THEOREM 4.
%,
normalization of
I(G)
L
such that, whenever h
There exists a
L
satisfies
I(,),
then
(61)
%G-a.e.
h =h
6 (G),
Further, for any nonzero measure
(62)
#0.
Let S and
PROOF.
be as in Theorem 3 and normalize
%,
such that the number
in (45) is unity.
Suppose first that h is bounded and continuous.
Choose A 91
aG.
(v) a (x)-w
lha (x)
and B(91
By (42) there exist f
lw
<
for some
(63)
for all x{A
2
such that B cA and
>_ 2
f fd% G
h(a)#V(a)
h (a), then
h(a)
such that, if w
Assume
B
f fd% G.
lwl
G\B
Consequently
G [h
f
(")a] "fd%Gt >_ BI w
2
a
d% G
f
G\B
(llhllo= +
II"II) fd%G
> 0.
(64)
Direct calculation shows
(,"),
(ga)^v
for all g
(LI(,)
(65)
ELl() (LI(G)
(66)
(LI(G)
with
and application of Fubini’s Theorem yields
f
d%
G
d%,
for all g
(65), (66) and Lemma 5 yield
Then
f
G
(v) a
f d%
f ha
G
which contradicts (64).
If
=/ g
G
f
G
G
"
(ha)^" f d%G= f(ha )^ dAw
d%w
f
h
".
Thus
G
h
a
f d%
G
((G) is nonzero, there exists a continuous function
pact support such that
f g(-t) d(t) # 0.
gig
C
with com-
Then the continuous function g,
G
nonzero at 0 and we may define
A
{x
S:
lg* U(-x)
>
Ig * U,0,’’!
2
By (42) there exist
f E C and
B
91
is
K. MCKENNON
456
such that BcA and
211g.
Ifd% G >
B
I
Ig * (o)
GXB
* 1.I,(0)
f dX
f d:k
G
Thus
/G g * (t)
Hence
f, g
f(t)
,
g
I
>
dXG(t)
2
B
j.
G
G\B
llg, II
f dX
> 0
G
is nonzero continuous and
II(f, g
. p)^ll
(g, p)^ll
Thus (61) implies that (f, g,
)v
-<
IIII 1
ll(g
,)^Iloo
Hence 0 # (f
f, g, P.
< oo.
, g, )^= (f, g)^
which establishes (62).
L
Now suppose that h
net given by
Then
lira
0
llm sup
xG
(61) that
L
6L
satisfying
I()
is arbitrary.
(43) of Theorem 3 and, for each index e, let
IIG- c," 111
Because each
I(G)
(g,h)
I(G) NL(G)
be the
go fl]lf
0 so
I/ [(y)
gc, * h
Let f
(y)]
e(y)
If
such that
j"
"-,,
G
dX
G
>
"
dXCy)I-- limll fi v- (g,h)^ll
ll(gm, h)
is continuous and
=g,h.
x,y
IIEII IIII
111
were not in
LI(G),
<,
+/-
(67)
follows from
there would exist
Ilhlll. Ilelloo
but (67) implies
J"G Ca" ,g
an absurdity.
and
,
that
dXGI =lira
Thus,
lGf g,h" dXGI -< llgcxll.n
v 6 LI(G)
the continuity of
=h.
Q.E.D.
Ilhll I
.
II11oo=llhlllll.elloo
Interchanging the roles of
and (61) imply that
"^
and
,
and of G
It now follows from (62)
FUNDAMENTAL THEORS OF HARMONIC ANALYSIS
457
REFERENCES
I.
GUICHARDET, A. Analyse Harmonlque Commmtative.
2.
HEWITT, Edwin and Ross, Kenneth, Abstract Harmonic Analysis, Vol. I,
Academic Press, New York; Sprfnger-Verlag, Berlin, 1963.
Dunod, Paris, 1968.
Abstract Harmonic A.alys.is. Vol. II, Springer-Verlag,
3o
New York and Berlin, 1970.
4.
NOVAK, David and McKennon, Kelly, Exponentials on Locally Compact Abelian
Groups, Proc. Am. Math. Soc., 83 (1981), 307-314.
5.
RAIKOV, D.A.
Invariant
Positive Definite Functions on Commutative Groups with an
Measure, Dokl. Aka. Mank SSSR, N.S. 28 (1940), 296-300.
6.
RUDIN, Walter, Fourier Analysis on Groups.
New York and London, 1962.
7.
VAN KAMPEN, E.R. Locally Bicompact Abelian Groups and their Character Groups,
Ann. of Math. (2) 36 (1935), 448-463.
Interscience Publishers,