Regularity Properties of Exponential Polynomials on Groups L

Acta ~lath. Hung. 45(1---2) (1985), 21--26. REGULARITY PROPERTIES OF EXPONENTIAL POLYNOMIALS ON GROUPS L. SZ15KELYHIDI (Debrecen)

I. Introduction

In a recent paper [13] we have studied regularity properties of polynomials on groups. The results extend some classical theorems and can be applied in the theory of functional equations [1], [8], [9], [10]. Besides polynomials, another important class for the applications is the class of exponential polynomials. The present paper is devo- ted to the study of regularity properties of exponential polynomials on Abelian groups. Concerning polynomials we shall use the same notations and terminology as in [13]. Concerning exponential polynomials one can find further investigations and results in [2], [3], [4], [5], [6], [71, [81, [9], [10], [111, [12]. If G is an Abelian group, then every homomorphism of G into the multiplicative group of nonzero complex numbers is called an exponential. If H is a complex vec- n torspace and a fimction f: G~H can be expressed in the form f= ~ mip~, where i=1 Pi: G-~H is a polynomial and mi: G-~C is an exponential, then.f is called an exponential polynomial. If further Imp[= 1 and p~ is constant for all i, then f is called a trigonometric polynomial. As in the case of polynomials, here we shall also use difference operators on G. Concerning the notations and definitions see [13]. We shall often use the following two lemmas which can be proved easily by induction: LEMMA 1.1. Let G be an Abelian group, H a complex vectorspace, in: G-+C an exponential, p: G-+H a function and f =mp. Then for all x, yCG we have

a~'f(x)=m(x) ~ (N)k (m(y)-I)N-kA~p(x+(N--k)Y)" k=O LEMMA 1.2. Let G be an Abelian group, H a complex vectorspace, A: G~H an n-additive symmetric function, q: G-~H a polynomial of degree at most n--l, and f=m(A(')+q). Then for all yCG there exists a polynomial qy: G~H of degree at most n-1 such that for all xE G we have agf(x) = m (x) [(m (y)- *)~a(") (x) + q, (x)]

(Here A (") denotes the diagonalization of A, that is the homogeneous polynomial of degree n generated by the n-additive symmetric function A.)

Acta Mathema~ica Hungarica 45, 1985 22 L. SZ~KELYHIDI

2. Zeros of exponential polynomials

THEOREM 2.1. Let G be an Abelian group, H a complex vectorspaee, mf : G~C an n exponential and Pi: G~H a polynomial (i=1, ..., n). If ~ mipi=O, then p~=0 (i=l, ..., n). i=1 PROOF. We prove the statement by induction on n. It is trivial for n = 1, hence we suppose that it is proved for n<=k- 1. Assume that n=k>-_2, and let p~---A(~",) +q~ where At: G",~H is an n-additive symmetric function with diagonafization A~("~), and q~: G~H is a polynomial of degree at most nrl (i=2,...,n). Further suppose that Px is of degree N. Obviously, it is enough to show that A 2("2)-n --u. Let yfi G such that ms(y) #ml(y). By Lemma 1.2 we have

11 A~ +xpa (x) + Z m, (x) ma (x) -~ [(m i (y) ma (y) -~ - 1)~ +~ A}",) (x) + qi,, (x)] = 0 i=2 where qz, y: G~H is a polynomial of degree at most ni-1. Then we obtain the equality mi(x)[(m~(y)m,(y) -1-1)N+~;4} ") (x)+ qi, y (x)] = 0, i=2 and by our assumption it follows for i=2, ..., n and xCG (m,(y) ml(y) -1-1)N +I A} ") (x) + q,,r(x) = O. Here qi, y is of degree at most n~-l, and by m~(y)ml(y)-l#l the first term for i=2 is of degree n~, hence our statement follows. We remark that by the above theorem the representation of exponential polynomials with different exponentials is unique. In the topological case:the above theorem has two generafizations concerning the zeros of exponential polynomials. Here we cite only the results, each of them is proved in [11]. THEOREM 2.2. Let G be a topological Abelian group which is generated by any neighborhood of zero and let H be a complex veetorspace. If an exponential polynomial f: G-~ H vanishes on some nonvoid open set, then it vanishes everywhere. THEOREM 2.3. Let G be a locally compact Abelian group which is generated by any neighborhood of zero and let H be a complex vectorspaee. If an exponential polynomial f: G~H vanishes on a measurable set of positive measure, then it vanishes everywhere.

3. Boundedness of exponential polynomials

THEOREM 3.1. Let G be an Abelian group, H a locally convex topological vectorspace, and f: G-~H an exponential polynomial. If f is bounded, then it is a trigonometric polynomial.

n PROOF. Let f= .~m, Pi, where m,; G-~C is an exponential, m,#mj for i=1 ' i#j, Pi: G~H is a polynomial and pf#0 (i,j=l, ..., n). We prove by induction

Acta Mathematiea Hungarlea 45, 1985.