Spectral synthesis via moment functions on hypergroups

Abstract

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.