Spectral synthesis via moment functions on hypergroups (Q2171908)

A hypergroup is roughly speaking a locally compact Hausdorff space which has enough structure so that a convolution on the corresponding vector space of Radon measures makes it a Banach algebra, with an identity element and a continuous involution. The hypergroup $X$ does not generally have an algebraic structure so algebraic operations are performed via the convolution algebra on $M(X)$. If $f$ is a complex valued function on $X$, \(f(x*y) = \delta_x * \delta_y (f)\) where \(\delta_x * \delta _y\) is a probability measure with compact support. In this paper the authors recall the definition of polynomial hypergroup and introduce necessary concepts that will ease the generalization from groups to hypergroups. The main result is the extension of the proof given in a previous paper [\textit{Ż.~Fechner} et al., Aequationes Math. 95, No.~6, 1281--1290 (2021; Zbl 1489.39034)] that characterizes the variety spanned by moment functions. Concretely, the authors establish that the Fourier algebra of a polynomial hypergroup in $d$ variables is the polynomial ring in $d$ variables. The main result generalizes to several variables the result on one variable stating that the $m$-sine functions on a variety in more than one dimension form a linear space of dimension at most one.