SECOND DEGREE POLYNOMIALS AND THE FUNDAMENTAL THEOREMS OF HARMONIC ANALYSIS

The concept of a second degree polynomial with nonzero subdegree is investigated 
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.

f(0) 0 for all f EP2(G,M); HOM(G,M) c P2(G,M).For fiG + M, define f*IG / M by f*(x) f(-x) Clearly f* EP2(G'M) for all f P2(G,M) A function fiG M is called even if f= f*.Write (4) (5)   for all xG.(6)   EP2(G) for the set of all even M has no element of order 2.
Thus (9) is equivalent to the evenness.
If fiG M satisfies f*=-f, it is said to be odd.Write OP2(G,M) for the set of all odd functions in P2(G,M).For f (P2(G,M) (7) implies f 60P2(G,M) f(2x) 2f(x) for all x EG. (I0) Putting x =2t+y and z =-2t-y in (2), one obtains 2f(y) + 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) 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. Define 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) and p HOM(ZmZ, C2).
For fiG-M and a(G the translation f IG-M is defined by f (x) f(x-a) for all x G. Thus the only translatlon-invariant functions are constants.A function fig M will be said to be translatlon-pseudo-invarlant (TPI) if, for each a E G, there exists f,a AFF(G,M) such that f e f'a + f.
3. 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.) For any such group G we write %G for a Haar measure on G, LP(G) for the LP-space relative to %G (p [I'] )' and (G) for the Banach space of bounded, complex, regular Borel measures on G and (G) for the set of continuous, positive-definite functions on G. Let T be the group of complex numbers of unit modulus and any LCA group topologically isomorphic to HOM(G, T).
The fundamental theorems of Abelian abstract harmonic analysis are as follows: (DUALITy THEOREM) There exists a continuous bihomomorphism e from to T such that, if for a G and b (UNIQUENESS THEOREM) defined by I Y f f(t)-tey d%G(y and G x f h(s) x e s d% (s), G then % can be normalized such that, whenever is in LI() for f LI(G), then (PLANCHEREL'S THEOREM) % can be normalized such that, for each 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 Algebras.It is based on the observation that, under the convolution operation , defined by f , h(x) I f(x-t) h (t)dt for all f, h 6L I(G) and x 6 G, 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 "wa.s first published in 1935 by van Kampen [7].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 LI(G) based on the structure theory (see [2] and [3] 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.The basic techniques are use of TPIfunctions and simple complex analysis.The 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 and its subgroup T. We define the complexificatlon G of G to be an abstract group isomorphic to HOM( ,T) containing G as a subgrou and extend o to a map on G such that, if for each a G a is defined by a I y a e y, then G a HOM(G,T) is an isomorphism. (31) In the special case G= T, we have Z and n T zo n z for all z and n Z. (32) As a second example, consider finite dimensional real Hilbert space E. Then G E and E can be taken any complex Hilbert space in which E is a real form--here e for all E 8 E (33) A function f which is the restriction to G of a function fl G such that fo is analytic for each 6 AFF(C,G), is said to be entlre--the Uniqueness Theorem for analytic functions implies that the function f is unique For f entire and a G we define the translation f of a f by The complexification , the concept of an entire function defined on C, and its translations are defined analogously.
We write LI(G) for the set of all entire functions f on G such that, for each AFF(,G) and > 0, sup{llfo(z)lll: Izl _< 6} < (where is the Ll-norm).In particular L I (G) c LI(G).LEMMA I. Let f L I(G) and a G be arbitrary.Then (35) PROOF.The function l Y a y can be written ll" __a where ll HOM(,]R+) and Then (35) implies that, for any simple closed contour y in , I If (x-(z)) d% GCx) dz < YG so it follows from Fubini's Theorem and Cauchy's Theorem that f f f(x-T(z)) d%G(X) dz f f f(x-T(z)) dz d%G(X) f0d%G=0.yG GY G Morera's Theorem now implies that the function FI zf f(x-T (z))d%G(X) is G analytic.For imaginary z, (z) is in HOM(,T) so (z) is in G--hence T(z) is in G as well and we have

HOM(,T). Define o HOM(,G) by letting, for
by the invariance of the Haar integral.The Uniqueness Theorem for analytic functions now implies that (37) holds for all z ( In particular F(0) G f f dA G But T(0) =a, so (36) holds.Q.E.D.
Relative to study of the Fourier Transformation, the utility of P2(G,T ) is dependent on the size of P2(G,T) LI(G).This size can vary, as is shown by Theorems 2 and 3 below.(38 Then f is in EP 2(E,T ) L I(E) and (where E is the dual of itself as in (33)).E PROOF.That f is in EP2(E follows from (17).
Let E and e be as in (33) and let be the conjugation of the HJlbert space k<Z z> leaving the elements of E fixed.Then the function fIE z is analytic so f is entire.Lemma implies that, for each y (E, llflll f k<t -iy,(t iy)> dE(t) f k<t't>e-21<In k y,t> k-<y,y> dE(t) E E Letting x--21nk "y, we have (39).Q.E.D.
LEMMA 3. Let K be a compact subgroup of G and h an element of P2(G,TC).
Then h(K) T.
In view of (21) there exist p EP2(G C T) and g 0P
From (13) we have g2 HOM(G,T), whence follows that g(K) 2  PROOF.Let G be an LCA group with a sequence S of compact subgroups such n that lira )tG(Sn)=' (such as a discrete group which contradicts Burnside's conjecture).Then, for h P2(G,T), Lemma 3 implies that h(S n) c T for each n.
Thus llhll I _> limn fS lhl dG=limn G(Sn)--" Q.E.D. n LEMMA 4. Suppose that G is compact and that f is a linear combination of characters.Then f %G(G) f.
PROOF.Evidently we need only consider the case f b for b .Since the fact that (b)-a =(-a) implies (-a) Consequently is %G(G) times the characteristic function of the singleton b.
Similarly v must be %G(G) .Q.E.D. Let be the directed set of open neighborhoods of 0 in G.
THEOREM 3.There is an open subgroup S of G, a set c L I(G) and a surjection x (A,m)f(A,m) satisfying the following properties.
flG\S =0 and fls is in the convex hull of P2(S,T ) for all f (40) f(0) =I and 1 f >_ 0 for all f (41) For each A there exists B such that BoA and (42) PROOF.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 hA =-1 + n y + (_y) 2 yEF(A) Then h A satisfies hA(O) i, 1 >_ h A >_ O, and hA(X) < 1 for all x E MA M- For each m]N, define f(A'm) G /JR + by f(A'M) IGS 0 and f(A'm)(x+y) =e-m<x'x> hA(Y) for xE, yM.
That (40) holds is evident from Lemma 2, that HOM(S,T) c P2(S,T), and the construction of f(A,m) Let be the set of all such functions f(A,m) That (41) holds is obvious.
PROOF.Let S and be as in Theorem 3 and normalize %, such that the number in (45) is unity.
Suppose first that h is bounded and continuous.Assume h(a)#V(a) for some aG.Choose A 91 such that, if w h(a) h (a), then lh (x) (v) (x)-w < lw for all x{A (63) Direct calculation shows (,"), (ga)^v for all g (LI(G) with (LI(,) and application of Fubini's Theorem yields f d% G =/ g d%, for all g ELl() (LI(G) G (66) Then (65), (66) and Lemma 5 yield f (v) f d% G G f (ha)^" f d%G= f(h )^d Aw G a a f h " d%w f h f d% G a G a G which contradicts (64).Thus h ".
If ((G) is nonzero, there exists a continuous function gig C with com- pact support such that f g(-t) d(t) # 0. Then the continuous function g, is G nonzero at 0 and we may define A {x S: lg* U(-x) > Ig * U,0,''!By (42) there exist f E C and B 91 2 odd is clear.That p is in P2(Z Z, C2) is evident from the following.If x=mn, if y= 9 and z =MN, then m,n even others arbitrary p(y+z) 0 p(y) ..p(z)p(y)

SBH
+ f(x) fa+b(X) fa(-b+x) g(a) + a(-b+x)+ f(-b+x) g(a) + a(-b) + Ha(x) + g(b) +b(x) + f(x), whence follows that a+b g(a+b) + N g(a) + g(b) + Ha(b) + n a + n b (24) Since the left side of (24) is symmetric in a and b, the right side is as well Thus a b (b) (a) for all a,b G.Consequently the function IGxG (x,y) X(y) is symmetric and is, in fact, in ) + g(b) + (a,b) for all a,b G.
ll G each z , o(z) be the element of such that o

LEMMA 2 .
Let E be finite dimensional real Hilbert space and k ]0,I[.Define f E C by f(x) k <x'x> for all x E.

Furthermore
we have that p#( SBIt(G TI) subgroup of T and so a subgroup of T. for x,y G Thus, for each x (G, p#(x,K) is a compact ) c p(K) g(K) c T" TfT.Q.E.D.

THEOREM 2 .
There are LCA groups G such that L I(G) P2 (G T) is void an open subset of {x6M:hA(X)> 1/2} and r' a positive number such that B E(r') + D is a subset of A. Then (54) yields m-n/2 I e -<s,s> dA E(s) EkE(mr') which proves (42).