Spectral synthesis via moment functions on 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.