### A positivity result and normalization of positive convolution structures.

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We introduce the analogue of Dunkl processes in the case of an affine root system of type ${\tilde{A}}_{1}$. The construction of the affine Dunkl process is achieved by a skew-product decomposition by means of its radial part and a jump process on the affine Weyl group, where the radial part of the affine Dunkl process is given by a Gaussian process on the ultraspherical hypergroup $[0,1]$. We prove that the affine Dunkl process is a càdlàg Markov process as well as a local martingale, study its jumps, and give a martingale...

We investigate amenability and weak amenability of the l¹-algebra of polynomial hypergroups. We derive conditions for (weak) amenability adapted to polynomial hypergroups and show that these conditions are often not satisfied. However, we prove amenability for the hypergroup induced by the Chebyshev polynomials of the first kind.

Let $X$ be a hypergroup. In this paper, we define a locally convex topology $\beta $ on $L\left(X\right)$ such that ${(L\left(X\right),\beta )}^{*}$ with the strong topology can be identified with a Banach subspace of $L{\left(X\right)}^{*}$. We prove that if $X$ has a Haar measure, then the dual to this subspace is ${L}_{C}{\left(X\right)}^{**}{=cl\{F\in L\left(X\right)}^{**};F$ has compact carrier}. Moreover, we study the operators on $L{\left(X\right)}^{*}$ and ${L}_{0}^{\infty}\left(X\right)$ which commute with translations and convolutions. We prove, among other things, that if $wap\left(L\right(X\left)\right)$ is left stationary, then there is a weakly compact operator $T$ on $L{\left(X\right)}^{*}$ which commutes with convolutions if and...

Mathematics Subject Classification: 42B35, 35L35, 35K35In this paper we study generalized Strichartz inequalities for the wave equation on the Laguerre hypergroup using generalized homogeneous Besov-Laguerre type spaces.

Let K be an ultraspherical hypergroup associated to a locally compact group G and a spherical projector π and let VN(K) denote the dual of the Fourier algebra A(K) corresponding to K. In this note, invariant means on VN(K) are defined and studied. We show that the set of invariant means on VN(K) is nonempty. Also, we prove that, if H is an open subhypergroup of K, then the number of invariant means on VN(H) is equal to the number of invariant means on VN(K). We also show that a unique topological...

We investigate the local Hardy spaces ${h}^{p}$ on Chébli-Trimèche hypergroups, and establish the equivalence of various characterizations of these in terms of maximal functions and atomic decomposition.