Spectral Synthesis Via Moment Functions on Hypergroups

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In particular, we show that if X is a polynomial hypergroup in d variables, then every exponential monomial is the linear combination of generalized moment functions, however, there are exponential monomials, whose variety contains more than one linearly independent sine functions, if d ą 1. A hypergroup is a locally compact Hausdorff space X equipped with an involution and a convolution operation defined on the space of all bounded complex regular measures on X. For the formal definition, historical background and basic facts about hypergroups we refer to [2]. In this paper X denotes a commutative hypergroup with identity element o, involution , and convolution ˚. For each function f on X we define the function f by fpxq“ fpxq. q Given x in X we denote the point mass with support the singleton txu by δx. q q The convolution δx ˚ δy is a compactly supportedq probability measure on X, and for each continuous function h: X Ñ C the integral

hptqdpδx ˚ δyqptq żX will be denoted by hpx ˚ yq. Given y in X the function x ÞÑ hpx ˚ yq is the translate of h by y. A set of continuous complex valued functions on X is called translation invariant, if it contains all translates of its elements. A linear translation invariant subspace of all continuous complex valued functions is called a variety, if it is closed with respect to uniform convergence on compact sets. The smallest variety containing the given function h is called the variety of h, and is denoted by τphq. Clearly, it is the intersection of all varieties including h. A continuous complex valued function is called an exponential polynomial, if its variety is finite dimensional. An exponential polynomial is called an exponential monomial, if its variety is indecomposable, that is, it is not the sum of two proper subvarieties. The simplest nonzero exponential polynomial is the one having one dimensional variety: it consists of all constant multiples of a nonzero continuous function. Clearly, it is an exponential monomial. If we normalize that function by taking 1 at o, then we have the concept of an exponential. Recall that m is an exponential on X if it is a non-identically zero continuous complex-valued function satisfying mpx ˚ yq “ mpxqmpyq for each x, y in X. Exponential monomials and polynomials on commutative hypergroups have been introduced and characterized in [9, 10, 3]. In the study of exponential polynomials the basic tool is the modified difference (see in [9]). Here we briefly recall the notion of modified difference. For a given function f in CpXq, y in X and an exponential m: X Ñ C we define the modified difference:

∆m;y ˚ fpxq“ fpx ˚ yq´ mpyqfpxq

for all x in X. For y1,...,yN`1 in X we deﬁne

∆m;y1...,yN`1 ˚ fpxq“ ∆m;y1...,yN ˚r∆m; yN`1 ˚ fpxqs for all x in X. The following characterization theorem of exponential monomials is proved in [3, Corollary 2.7]. 3

Theorem 1.1. Let X be a commutative hypergroup. The continuous function f : X Ñ C is an exponential monomial if and only if there exists an exponential m and a natural number n such that

∆m;y1,y2,...,yn`1 ˚ f “ 0 holds for each y1,y2,...,yn`1 in X. If f is nonzero, then m is uniquely determined, and f is called an m-exponential monomial, and its degree is the smallest n satisfying the property in Theorem 1.1. The degree of each exponential function is zero: in fact, every nonzero exponential monomial of degree zero is a constant multiple of an exponential. For exponential monomials of first degree sine functions provide an example: given an exponential m on X the continuous function s : X Ñ C is called an m-sine function, if spx ˚ yq“ spxqmpyq` spyqmpxq holds for each x, y in X. If s is nonzero, then m is uniquely determined, and its degree is 1. Important examples for exponential polynomials are provided by the moment functions. Let X be a commutative hypergroup, r a positive integer, and for each r multi-index α in N , let fα : X Ñ C be continuous function, such that fα ‰ 0 for |α|“ 0. We say that pfαqαPNr is a generalized moment sequence of rank r, if α (1.1) f px ˚ yq“ f pxqf pyq α β β α´β βďα ÿ ˆ ˙ holds whenever x, y are in X (see [4]). It is obvious that the variety of fα is finite dimensional: in fact, every translate of fα is a linear combination of the finitely many functions fβ with β ď α, by equation (1.1). The functions in a generalized moment function sequence are called generalized moment functions. In this paper we shall omit the ”generalized” adjective: we simply use the terms moment function sequence and moment function. The order of fα in the above moment function sequence is defined as |α|. It follows that moment functions are exponential polynomials. The special case of rank 1 moment sequences leads to the following system of functional equations: k k f px ˚ yq“ f pxqf pyq k j j k´j j“0 ÿ ˆ ˙ for all k “ 0, 1,...,N and for each x, y in X.

Observe that in every moment function sequence the unique function fp0,0,...,0q is an exponential on the hypergroup X. In this case we say that the exponential m “ fp0,0,...,0q generates the given moment function sequence, and that the moment functions in this sequence correspond to m.We may also say that the moment functions in the given moment sequence are m-moment functions. It is also easy to see, that in every moment function sequence fα with |α| “ 1 is an m-sine function on X, where m “ fp0,0,...,0q. The main result in [4] is that moment sequences of rank r can be described by using Bell polynomials if the underlying hypergroup is an Abelian group. The point 4 ZYWILLA˙ FECHNER, ESZTER GSELMANN, AND LASZL´ O´ SZEKELYHIDI´ is that in this case the situation concerning (1.1) can be reduced to the case where the generating exponential is the identically 1 function and the problem reduces to a problem about polynomials of additive functions. Unfortunately, if X is not a group, then such a reduction is not available, in general. The following result is important. Theorem 1.2. Let X be a commutative hypergroup. Then the variety of a nonzero moment function contains exactly one exponential: the generating exponential function. Proof. In the moment function sequence in (1.1) we show that the only exponential in the variety of f “ fα is m “ fp0,0,...,0q. First we show that f is an m-exponential monomial of degree at most |α|. We prove by induction on N “ |α|, and the statement is obviously true for N “ 0. Assuming that the statement holds for |α|ď N, we prove it for |α|“ N ` 1. By equation (1.1), we have

(1.2) ∆m;y ˚ fαpxq“ fαpx ˚ yq´ mpyqfαpxq“ α f px ˚ yq´ f pyqf pxq“ f pxqf pyq. α p0,0,...,0q α β β α´β βăα ÿ ˆ ˙ for each x, y in X. The right side, as a function of x, is an exponential monomial of degree at most N “ |α|´ 1, corresponding to the exponential m, by the induction hypothesis. It follows that

∆m;y1,y2,...,yN`1,y ˚ fαpxq“ ∆m;y1,y2,...,yN`1 ˚r∆m;y ˚ fαspxq“ α r∆ 1 N 1 ˚ f pxqsf pyq“ 0, β m;y ,...,y ` β α´β βăα ÿ ˆ ˙ which proves our ﬁrst statement. To prove that m is the only exponential in the variety of f we know, from the results in [11], that the annihilator Ann τpmq of the variety τpmq of m is a closed maximal ideal Mm in McpXq, the space of all compactly supported complex Borel measures on X, in which the measures ∆m;y span a dense subspace, and τpmqĎ τpfq, hence Ann τpfqĎ Mm. On the other hand, by the above equation, N`2 Mm Ď Ann τpfq, as the product of any N ` 2 generators of a dense subspace of Mm annihilates the function f. It follows that, if an exponential m1 is in τpfq, then clearly τpm1q Ď τpfq, hence Ann τpfqĎ Ann τpm1q. We conclude that N`2 Mm Ď Ann τpfqĎ Ann τpm1q.

As Ann τpm1q is a maximal ideal, hence it is a prime ideal, as well, consequently we have Mm Ď Ann τpm1q. By maximality, it follows that we have τpmq“ τpm1q, which immediately implies m1 “ m. This theorem can be formulated in the way that every m-moment function is an m-exponential monomial. Exponential monomials are the basic building blocks of spectral synthesis. We say that spectral analysis holds for a variety, if every nonzero subvariety in the variety contains a nonzero exponential monomial. We say that a variety is synthe- sizable if all exponential monomials in the variety span a dense subspace. We say 5 that spectral synthesis holds for a variety if every subvariety of it is synthesisable. It is obvious that spectral synthesis for a variety implies spectral analysis for it. If spectral analysis holds for every variaty on X, then we say that spectral analysis holds on X. If every variety on X is synthesisable, then we say that spectral synthesis holds on X. Clearly, on every commutative hypergroup, spectral synthesis holds for finite dimensional varieties. In the light of these definitions our basic problem about the relation between exponential monomials and moment functions can be reformulated: is it true that every finite dimensional variety is spanned by moment functions? If so, then spectral synthesis is rather based on moment functions, not on exponential monomials. Obviously, it is enough to consider indecomposable varieties, and even we may assume that the variety in question is the variety of an exponential monomial. Our result in [5, Theorem 2.1] says that if the linear space of all m-sine functions in the variety of an m-exponential monomial is (at most) one dimensional, then this variety is spanned by moment functions – obviously, associated with m. In the subsequent paragraphs we show that this may happen also in cases where the m-sine functions span a more than one dimensional subspace in the variety.

2. Polynomial hypergroups In this section we recall the definition of a class of hypergroups which will be used in the sequel. The following definition is taken from [2, Chapter 3, Section 3.1, p. 133]. Let d d denote a positive integer, let Pn denote the set of polynomials in on C of degree at most n. Finally, let π denote a probability measure on Cd, and we assume that the commutative hypergroup X with convolution ˚, involution , and identity o satisfies d the following properties: for each x in X there exists a polynomial Qx on C such that we have: q (P1) For each nonnegative integer n, the set Xn “tx : x P X,Qx P Pnu is a basis of the linear space Pn. (P2) We have Qxp1, 1,..., 1q“ 1 for each x in X. (P3) For each x in X, we have Qxpzq“ Qxpzq whenever z is in supp π. (P4) For each x, y, w in X, we have q

QxQyQw dπ ě 0. Cd ż In fact, the properties (P1), (P2), (P3), (P4) are not independent – for the details see [2, Proposition 3.1.4]. Property (P4) expresses that the polynomials Qx for x in X are orthogonal with respect to the measure π.

In this case, by property (P1), for each x, y in X the polynomial QxQy admits a unique representation

(2.1) QxQy “ cpx, y, wqQw wPX ÿ with some complex numbers cpx, y, wq. The formula (2.1) is called linearization formula, and the numbers cpx, y, wq are called linearization coeﬃcients. The hypergroup X with convolution ˚ is called a polynomial hypergroup (in d variables) if there exists a family tQx : x P Xu of polynomials satisfying the above 6 ZYWILLA˙ FECHNER, ESZTER GSELMANN, AND LASZL´ O´ SZEKELYHIDI´ property and

δx ˚ δy “ cpx, y, wqδw wPX ÿ for each x, y in X. In this case we say that the polynomial hypergroup X is associated with the family of polynomials tQx : x P Xu. The choice d “ 1 is the case of polynomial hypergroups in one variable. In this case we may suppose that X “ N, the set of natural numbers, and the family of polynomials tQx : x P Xu is in fact a sequence of polynomials pPnqnPN, where Pn is of degree n and we always assume that P0 “ 1. Suppose that there exists a Borel measure on a finite interval such that the sequence pPnqnPN is orthogonal with respect to this measure. In this case it turns out that the linearization coefficients cpk,l,nq are zero unless |k ´ l|ď n ď k ` l. Hence the linearization formula has the form k`l (2.2) PkPl “ cpk,l,nqPn. l k l “|ÿ´ | In this situation, the sequence pPnqnPN generates a polynomial hypergroup if and only if the linearization coefficients are nonnegative.

As a particular example we consider the sequence pTnqnPN of Chebyshev polynomials of the first kind defined by the recursion T0pxq“ 1, T1pxq“ x and 1 xT pxq“ rT 1pxq` T 1pxqs n 2 n´ n` for n “ 1, 2,... In this case it is easy to check that the linearization coefficients are nonnegative and the resulting hypergroup is the Chebyshev hypergroup. For more about polynomial hypergroups in one variable see also [2, 8]. Based on [2, 3.1.3. (iii)], this construction can be generalized for dimension d ą 1. Indeed, if Kp1q and Kp2q are polynomial hypergroups, then Kp1q ˆ Kp2q can be made into a polynomial hypergroup, as well. The collection

p1q p2q Qx b Qy : px, yqP K ˆ K ! ) of polynomials on Cd1`d2 defines a hypergroup structure on Kp1q ˆ Kp2q, where p1q p1q p2q p2q Qx : x P K and Qy : y P K denote the collections of polynomials on !Cd1 and Cd2 defining) K!p1q and Kp2q, respectively.) Thus, the notion of Chebyshev hypergroups can also be extended for d ě 2. For example, in case d “ 2, after the above construction, a hypergroup arises that can also be defined with the following recursion. Let T0,0px, yq“ 1, T1,0px, yq“ x, T0,1px, yq“ y, T1,1px, yq“ xy, for each nonnegative integers k,n, let

Tkpxq, if n “ 0 Tk,npx, yq“ #Tnpyq, if k “ 0, furthermore, for each positive integers k,n with k ` n ě 2,

(2.3) xyTk,npx, yq“ 1 rT 1 1px, yq` T 1 1px, yq` T 1 1px, yq` T 1 1px, yqs. 4 k´ ,n´ k` ,n´ k´ ,n` k` ,n` 7

Clearly, this is satisfied by Tk,npx, yq“ Tkpxq ¨ Tnpyq, which leads to a two dimensional generalization of the Chebyshev hypergroup on the basic set X “ N ˆ N. From the above recursion formulas we can derive 1 T ¨ T “ rT ` T ` T ` T s, k,l m,n 4 |k´m|,|lń| |k´m|,l`n k`m,|lń| k`m,l`n which provides us the convolution of point masses in X: 1 (2.4) δ ˚ δ “ rδ ` δ ` δ ` δ s k,l m,n 4 |k´m|,|lń| |k´m|,l`n k`m,|lń| k`m,l`n whenever k,l,m,n are natural numbers. The involution on X is the identity mapping, and the identity is δ0,0. Exponential functions on X are described in the following theorem (see [2, Propo- sition 3.1.2], [8, Theorem 3.1]). Theorem 2.1. The function M : N ˆ N Ñ C is an exponential on X if and only if there exist complex numbers λ, µ such that

(2.5) Mpx, yq“ TxpλqTypµq holds whenever px, yq are in X. The correspondence between the exponentials M and the pairs pλ, µq is one-to-one. Proof. If M has the given form with some complex numbers λ, µ, then for each natural numbers k,l,m,n we have

Mrδk,l ˚ δm,ns“ Mpu, vq dpδk,l ˚ δm,nqpu, vq“ żX 1 rT pλqT pµq` T pλqTl npµq` Tk mpλqT pµq` Tk mpλqTl npµqs “ 4 |k´m| |l´n| |k´m| ` ` |l´n| ` ` Tk,lpλqTm,npµq“ TkpλqTlpµq ¨ TmpλqTnpµq“ Mpδk,lqMpδm,nq, further Mpδ0,0q “ T0pλqT0pµq “ 1, hence M is an exponential on the hypergroup X. Conversely, suppose that M : N ˆ N Ñ C is an exponential on X, that is, we have

(2.6) Mrδk,l ˚ δm,ns“ Mpδk,lqMpδm,nq for each natural numbers k,l,m,n, further Mpδ0,0q“ 1. We deﬁne the function

fpm,nq“ Mpδm,nq whenever m,n are in N. Then f satisfies (2.7) fpk,lqfpm, nq“ 1 rMpδ q` Mpδ q` Mpδ q` Mpδk m,l nqs “ 4 |k´m|,|lń| |k´m|,l`n k`m,|lń| ` ` 1 rfp|k ´ m|, |l ´ n|q ` fp|k ´ m|,l ` nq` fpk ` m, |l ´ n|q ` fpk ` m,l ` nqs 4 for each k,l,m,n in N. Here we substitute l “ m “ 0 to get (2.8) fpk, 0qfp0,nq“ fpk,nq, and the substitution l “ n “ 0 in (2.7) gives 1 fpk, 0qfpm, 0q“ rfp|k ´ m|, 0q` fpk ` m, 0qs 2 8 ZYWILLA˙ FECHNER, ESZTER GSELMANN, AND LASZL´ O´ SZEKELYHIDI´ for all k,m in N. As fp0, 0q“ 1, the function k ÞÑ fpk, 0q is an exponential of the Chebyshev hypergroup (see [8]), hence fpk, 0q “ Tkpλq for each k in N with some complex number λ. Similarly, we have that fp0,nq“ Tnpµq holds for each n in N with some complex number µ. It follows, by (2.8), that

Mpδk,nq“ fpk,nq“ fpk, 0qfp0,nq“ TkpλqTnpµq, and our statement is proved. Clearly, the pair λ, µ is uniquely determined by the exponential M. Using this theorem it is reasonable to denote the exponential corresponding to 2 the pair pλ, µq in C by Mλ,µ.

Theorem 2.2. Let Mλ,µ : N ˆ N Ñ C be an exponential on X. The function S : N ˆ N Ñ C is an Mλ,µ-sine function on X if and only if there are complex numbers a,b such that 1 1 (2.9) Spx, yq“ aTxpλqTypµq` bTxpλqTypµq whenever x, y are in X.

Proof. The function S given in (2.2) is an Mλ,µ-sine function on X, as it is easy to check by simple calculation. For the converse we assume that S : N ˆ N Ñ C is an Mλ,µ-sine function on X. Then we have for each x,y,u,v in N:

(2.10) S px, yq ˚ pu, vq “ Spx, yqMλ,µpu, vq` Spu, vqMλ,µpx, yq. Using the form` of the exponentials˘ on X given in the previous Theorem 2.1 and substituting y “ u “ 0 we get

(2.11) Spx, vq“ Spx, 0qTvpµq` Sp0, vqTxpλq. Now we substitute y “ v “ 0 in (2.10) to get 1 (2.12) rSp|x ´ u|, 0q` Spx ` u, 0qs “ Spx, 0qT pλq` Spu, 0qT pλq. 2 u x

Using the fact that the function Mλ : x ÞÑ Txpλq is an exponential on the Chebyshev hypergroup (see [8]), we can write equation (2.12) in the form

(2.13) Spx ˚ u, 0q“ Spx, 0qMλpuq` Spu, 0qMλpxq, and this means that the function x ÞÑ Spx, 0q is an Mλ-sine function on the Cheby- shev hypergroup. It follows from [8, Theorem 2.5.], that d Spx, 0q“ a M pxq“ aT 1 pλq dλ λ x with some complex number a. Similarly, we have that d Sp0, vq“ b M pxq“ aT 1 pµq. dµ µ v Finally, substitution into (2.11) gives the statement.

It follows that the linear space of all Mλ,µ-sine functions on X is at most two 1 dimensional, it is spanned by the two functions px, yq ÞÑ TxpλqTypµq and px, yq ÞÑ 1 TxpλqTypµq. On the other hand, these two functions are linearly independent. Indeed, assume that 1 1 αTxpλqTypµq` βTxpλqTypµq“ 0 9 holds for some complex numbers α, β and for all x, y in X. Substituting x “ 0,y “ 1 1 1 and using T0pzq “ 1 and T1pzq “ z we have T0pλq “ 0 and T1pµq “ 1, hence we get β “ 0. Interchanging the role of x and y, we obtain α “ 0, hence the two 1 1 functions px, yq ÞÑ TxpλqTypµq and px, yq ÞÑ TxpλqTypµq are linearly independent – they form a basis of the linear space of all Mλ,µ-sine functions. We consider the variety τpϕq of an Mλ,µ-exponential monomial, which is not a linear combination of moment functions in τpϕq associated with the exponential Mλ,µ. From [5, Theorem 1 1 2.1] it follows, that the two functions px, yq ÞÑ TxpλqTypµq and px, yq ÞÑ TxpλqTypµq must belong to τpϕq, hence the linear space of all Mλ,µ-sine functions in τpϕq is two dimensional. We will show that still τpϕq is generated by moment functions. This would verify that the suﬃcient condition given in [5, Theorem 2.1] for that the variety of an exponential monomial is generated by moment functions is not necessary. In fact, we will show that, in general, on every polynomial hypergroup, every ﬁnite dimensional variety is spanned by moment functions. In particular, the variety of each exponential monomial is spanned by moment functions – it follows that every exponential monomial is a linear combination of moment functions. On the other hand, it is easy to see that on polynomial hyeprgroups in more than one variable there are exponential monomials whose variety contains more than one linearly independent sine function. We note that the formula for S in equation (2.9) can be written in the following form

Spx, yq“ aBλMλ,µpx, yq` bBµMλ,µpx, yq, which is a special case of the moment functions appear in Theorem 3.1 in the next section.

3. A class of moment functions on polynomial hypergroups Let X be a polynomial hypergroup in d variables associated with the family of polynomials tQx : x P Xu, and we always assume that Qo “ 1, where o is the identity of X. We introduce a special class of exponential monomials on X. Let P be a polynomial in d variables:

α P pξq“ aαξ , α N | ÿ|ď d where we use multi-index notation. This means that ξ “ pξ1, ξ2,...ξdq is in C , d α “ pα1, α2,...,αdq is in N , aα is a complex number, and

α α1 α2 αd |α|“ α1 ` α2 `¨¨¨` αd, ξ “ ξ1 ξ2 ¨ ¨ ¨ ξd . Then we write α P pBq “ aαB , α N | ÿ|ď where B “ pB1, B2,..., Bdq with the obvious meaning of the partial differential operators Bi. The differential operator P pBq is defined on the space of polynomials in d variables in the usual way: given the polynomial Q in the polynomial ring Crz1,z2,...,zds then

α1 α2 αd rP pBqQspξq“ aαrB1 B2 ¨¨¨Bd Qspξq α N | ÿ|ď 10 ZYWILLA˙ FECHNER, ESZTER GSELMANN, AND LASZL´ O´ SZEKELYHIDI´

Cd for each ξ in . If P is not identically zero, then we always assume that |α|“N |aα|ą 0, that is, the the total degree of the polynomial P is N – in this case we say that the diﬀerential operator P pBq is of order N. ř

The following result shows that the functions x ÞÑ P pBqQx are linear combinations of moment functions on X. Theorem 3.1. Let X be a polynomial hypergroup in d variables associated with d the family of polynomials tQx : x P Xu. Then, for each multi-index α in N , and d α for every λ in C , the function x ÞÑ rB Qxspλq is a moment function sequence of rank d associated with the exponential x ÞÑ Qx. Proof. Let cppx,y,tq denote the linearization coeﬃcients of the polynomial hypergroup X, that is Qx ¨ Qy “ cpx,y,tqQt tPX ÿα for each x, y in X. Further let fpxq“rB Qxspλq whenever x in X: then we have

α fpx ˚ yq“ fptq dpδx ˚ δyqptq“ rB Qtspλq dpδx ˚ δyqptq“ żX żX α α B r Qtpλq dpδx ˚ δyqptqs“B r cpx,y,tqQtpλqs “ X tPX ż ÿ α rBαpQ ¨ Q qspλq“ rBβQ spλq¨rBα´βQ spλq, x y β x y βďα ÿ ˆ ˙ which proves the statement.

The moment functions of the form x ÞÑBQxpλq play an important role in our work: our purpose is to show that in any variety the linear combinations of these moment functions span a dense subspace. In other words, spectral synthesis holds on any polynomial hypergroup even when we restrict ourselves to exponential polynomials merely of the form x ÞÑ P pBqQxpλq. This will be proved in the subsequent paragraphs.

4. The Fourier–Laplace transform In what follows we shall use the Fourier–Laplace transform on commutative hypergroups. Here we shortly summarize the basic concepts and results. Let X be a commutative hypergroup and let CpXq denote the linear space of all complex valued continuous functions on X. Equipped with the uniform convergence on compact sets CpXq is a locally convex topological vector space. If, for instance, X is discrete, then CpXq is the linear space of all complex valued functions on X and the topology on CpXq is the topology of pointwise convergence. The topological dual space of CpXq can be identiﬁed with the space of all compactly supported complex Borel measures on X, denoted by McpXq. The identiﬁ- cation depends on the Riesz Representation Theorem (see e.g. [6, Theorem 6.19]): every continuous linear functional Λ on CpXq can be represented in the form

Λpfq“ f dµ żX whenever f is in CpXq, where µ is a uniquely determined measure in McpXq, depending on Λ only. Clearly, if X is discrete, then McpXq is the linear space of all 11

finitely supported complex valued functions on X. For each measure µ in McpXq we define the measure µ by f dµ “ f dµ q X X ż ż whenever f is in CpXq. q q The convolution in X induces an algebra structure on McpXq in the following manner. Given two measures µ,ν in McpXq their convolution is introduced as the measure µ ˚ ν defined on CpXq by the formula

f dpµ ˚ νq“ fpx ˚ yq dµpxq dνpyq, żX żX żX whenever f is in CpXq. We recall that fpx ˚ yq stands for the integral

f dpδx ˚ δyq“ fptq dpδx ˚ δyqptq, żX żX hence the above formula can be written in the more detailed form as

f dpµ ˚ νq“ fptq dpδx ˚ δyqptq dµpxq dνpyq, żX żX żX żX whenever f is in CpXq. As all the measures µ,ν,µ ˚ ν are compactly supported, this integral exists for each continuous function f.

It is easy to check that the linear space McpXq is a commutative algebra with the convolution of measures. We call McpXq the measure algebra of the hypergroup X. If X is discrete, then it is usually called the hypergroup algebra of X – imitating the terminology used in group theory.

The Fourier–Laplace transform on McpXq is deﬁned as follows: for each measure µ in McpXq and for each exponential m on X we let

µpmq“ m dµ. żX E C E Then µ : pXq Ñ is a complexp valued functionq deﬁned on the set pXq of all exponentials on X. Clearly, the mapping µ ÞÑ µ is linear, and it is also an algebra homomorphism,p as it is shown by the following simple calculation: p pµ ˚ νq pmq“ m dpµ ˚ νq“ mpx ˚ yq dµpxq dνpyq“ żX żX żX p q q “ mpxqmpyq dµpxq dνpyq“ mpxq dµpxq ¨ mpyq dνpyq“ µpmq ¨ νpmq, żX żX żX żX which holds for each µ,ν in M pXq and for every exponential m on X. q q c q q p p From this property it follows that the set of all Fourier–Laplace transforms on EpXq form an algebra, which is called the Fourier algebra of X, denoted by ApXq. In fact, by one of the most important properties of the Fourier–Laplace transform it follows, that the Fourier algebra is isomorphic to the measure algebra, which is expressed by the following theorem (see e.g. [2, 2.2.4 Theorem]). Theorem 4.1. Let X be a commutative hypergroup and let µ be in the measure algebra McpXq. If µpmq“ 0 for each exponential m on X, then µ “ 0.

p 12 ZYWILLA˙ FECHNER, ESZTER GSELMANN, AND LASZL´ O´ SZEKELYHIDI´

For our purposes it will be necessary to describe the Fourier algebra of polynomial hypergroups. This will be done in the subsequent paragraphs.

Let X be a polynomial hypergroup in d variables generated by the family tQx : x P Xu of polynomials. We may assume that Qo “ 1, where o is the identity of X. It is known that the function m : X Ñ C is an exponential on X if and only if there exists a λ in Cd such that

mpxq“ Qxpλq holds for each x in X. As λ is obviously uniquely determined by m, hence the set EpXq of all exponentials on X can be identiﬁed with Cd. Consequently, the Fourier–Laplace transform of each measure in McpXq is a complex valued function on Cd:

µpλq“ Qxpλq dµpxq. żX In fact, the integral is a ﬁnite sum, which implies that µ is a complex polyno- p mial in d variables. Conversely, let P be any polynomial in the polynomial ring C rz1,z2,...,zds. Then, by deﬁnition, P has a representationp of the form n

P pzq“ ckQxk pzq k“1 ÿ with some elements xk in X and complex numbers ck for k “ 1, 2,...,n. We have

δxk pzq“ Qxpzq dδxk pxq“ Qxk pzq, żX hence p n n n

P “ ckQxk “ ckδxk “ ckδxk . k“1 k“1 k“1 ÿ ÿ ` ÿ ˘ n M p Here µ “ k“1 ckδxk is in cpXq, hence we have proved thatp each polynomial in Crz1,z2,...,zds is in ApXq. In other words, the Fourier algebra of each polynomial hypergroupř in d variables is the polynomial ring in d variables.

5. Spectral synthesis via moment functions We shall use the following basic result (see [7], [8, Theorem 6.9]). Theorem 5.1. (Ehrenpreis–Palamodov Theorem) Let I be a primary ideal in the polynomial ring Crz1,z2,...,zds, and let V denote the set of all common zeros of all polynomials in I. Then there exists a positive integer t such that, for i “ 1, 2,...,t there exist diﬀerential operators with polynomial coeﬃcients of the form

j1 j2 jd Aipz, Bq “ ci,j pj pz1,z2,...,zdqB1 B2 ¨¨¨Bd j ÿ for i “ 1, 2,...,t with the following property: a polynomial p in Crz1,z2,...,zds lies in the ideal I if and only if the result applying Aipz, Bq to f vanishes on V for i “ 1, 2,...,t. The following result can be found in [8, Theorem 6.10], but for the sake of completeness we present it together with its proof. 13

Theorem 5.2. Let X be a polynomial hypergroup in d variables, and let V be a proper variety on X. Then the functions of the form x ÞÑ P pBqQxpλq in V , where P d is a polynomial in d variables, and λ is in C such that the exponential x ÞÑ Qxpλq is in V , span a dense subspace in V . Proof. We use the fact that, for each variety on X, if V K denotes the orthogonal M complement of V , that is, the set of all measures µ in cpXq such that X f dµ “ 0 for each f in V , then V K is an ideal in M pXq. Similarly, for each ideal I, the c ş annihilator IK is the set of all functions in CpXq such that f dµ “ 0 for each µ in I, and it is a variety in CpXq. In addition, we have V KK “ V . (see e.g. [11, ş Theorem 5, Theorem 16]). In our case, in particular, V K is a proper ideal. By the Noether–Lasker Theorem, ([1, Theorem 7.13]) I is an intersection of (ﬁnitely many) primary ideals. It follows from the Ehrenpreis–Palamodov Theorem 5.1 that, for each λ in V there is a set Pλ of polynomials such that the measure µ in McpXq annihilates V if and only if

rP pBqµspλq“ rP pBqQxspλq dµpxq“ 0 żX holds for each λ in V and P in P . In other words, all exponential polynomials p λ x ÞÑ rP pBqQxspλq with x ÞÑ Qxpλq in V and P in Pλ belong to V , and their linear hull is dense in V . From this theorem we infer that every exponential monomial is a linear combination of moment functions. Corollary 5.3. Let X be a polynomial hypergroup. Then every exponential polynomial on X is a linear combination of moment functions contained in its variety. Corollary 5.4. Let X be a polynomial hypergroup in d variables. Then every exponential polynomial on X has the form x ÞÑ P pBqQxpλq with some complex polynomial P in d variables and some complex number λ. Proof. If f is an exponential monomial, then it has a finite dimensional variety V . As P pBq is a linear combination of differential operators of the form Bα, and, by α Theorem 3.1, x ÞÑ rB Qxspλq is a moment function, the previous theorem implies that the linear combinations of all exponential monomials in V span a dense subspace. By finite dimensionality, it means that V is the linear span of all moment functions included in V , hence f is a linear combination of moment functions. References

[1] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. [2] W. R. Bloom and H. Heyer, Harmonic analysis of probability measures on hypergroups, de Gruyter Studies in Mathematics, vol. 20, Walter de Gruyter & Co., Berlin, 1995. [3] Z.˙ Fechner and L. Székelyhidi, Finite dimensional varieties on hypergroups, Aequationes Math., doi.org/10.1007/s00010-021-00777-y [4] Z.˙ Fechner, E. Gselmann and L. Székelyhidi, Moment functions on groups, arXiv:2010.00208 [math.CA], 2020. [5] Z.˙ Fechner, E. Gselmann and L. Székelyhidi, Moment functions and exponential monomials on commutative hypergroups, to appear [6] W. Rudin, Real and complex analysis, McGraw-Hill Book Co., New York, third edition, 1987. [7] B. Sturmfels, Solving systems of polynomial equations, volume 97 of CBMS Regional Con- ference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2002. 14 ZYWILLA˙ FECHNER, ESZTER GSELMANN, AND LASZL´ O´ SZEKELYHIDI´

[8] L. Székelyhidi, Functional Equations on Hypergroups, World Scientific Publishing Co. Pte. Ltd., New Jersey, London, 2013. [9] L. Székelyhidi, Exponential polynomials on commutative hypergroups, Arch. Math (Basel), 101 (4), 341–347, 2013. [10] L. Székelyhidi, Characterization of exponential polynomials on commutative hypergroups, Ann. Funct. Anal., 5 (2), 53–60, 2014. [11] L. Székelyhidi, Spherical spectral synthesis on hypergroups, Acta Math. Hungar., 163 (1), 247–275, 2021.

Institute of Mathematics,, Lodz University of Technology,, 90-924L od´ ´z,, ul. Wolcza´ nska´ 215, Poland, ORCID 0000-0001-7412-6544 Email address: [email protected]

University of Debrecen,, H-4002 Debrecen,, P.O.Box: 400, Hungary Email address: [email protected]

University of Debrecen,, H-4002 Debrecen,, P.O.Box: 400, Hungary, ORCID: 0000- 0001-8078-6426 Email address: [email protected], [email protected]