Spectral synthesis via moment functions on hypergroups

Abstract

In this paper, we continue the discussion about relations between exponential polynomials and generalized moment 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 the 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.