Spectral synthesis via moment functions on hypergroups

TL;DR

This paper explores the relationship between exponential polynomials and moment functions on hypergroups, demonstrating that all exponential polynomials can be expressed as linear combinations of moment functions, extending previous results.

Contribution

It generalizes prior work by showing that varieties spanned by exponential functions can be represented by moment functions even when sine functions span more than one dimension.

Findings

Fourier algebra of polynomial hypergroups is a polynomial ring.

Every exponential polynomial on these hypergroups is a linear combination of moment functions.

The paper characterizes exponentials and sine functions on polynomial hypergroups.

Abstract

In this paper we continue the discussion about relations between exponential polynomials and generalized moment generating functions on a commutative hypergroup. We are interested in the following problem: is it true that every finite dimensional variety is spanned by moment functions? Let $m$ be an exponential on $X$. In our former paper we have proved 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 generated by $m$. In this paper we show that this may happen also in cases where the $m$-sine functions span a more than one dimensional subspace in the variety. We recall the notion of a polynomial hypergroup in $d$ variables, describe exponentials on it and give the characterization of the so called $m$-sine functions. Next we show that the Fourier algebra of a polynomial hypergroup in $d$ variables is the polynomial ring in $d$ variables. Finally, using Ehrenpreis--Palamodov Theorem we show that every exponential polynomial on the polynomial hypergroup in $d$ variables is a linear combination of moment functions contained in its variety.