Classical finite association schemes lead to finite-dimensional algebras which are generated by finitely many stochastic matrices. Moreover, there exist associated finite hypergroups. The notion of classical discrete association schemes can be easily extended to the possibly infinite case. Moreover, this notion can be relaxed slightly by using suitably deformed families of stochastic matrices by skipping the integrality conditions. This leads to a larger class of examples which are again associated with discrete hypergroups. In this paper we propose a topological generalization of association schemes by using a locally compact basis space $X$ and a family of Markov-kernels on $X$ indexed by some locally compact space $D$ where the supports of the associated probability measures satisfy some partition property. These objects, called continuous association schemes, will be related to hypergroup structures on $D$. We study some basic results for this notion and present several classes of examples. It turns out that, for a given commutative hypergroup, the existence of a related continuous association scheme implies that the hypergroup has many features of a double coset hypergroup. We, in particular, show that commutative hypergroups, which are associated with commutative continuous association schemes, carry dual positive product formulas for the characters. On the other hand, we prove some rigidity results in particular in the compact case which say that for given spaces $X,D$ there are only a few continuous association schemes.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}K$ be a locally compact hypergroup endowed with a left Haar measure and let $L^1(K)$ be the usual Lebesgue space of $K$ with respect to the left Haar measure. We investigate some properties of $L^1(K)$ under a locally convex topology $\beta ^1$. Among other things, the semireflexivity of $(L^1(K), \beta ^1)$ and of sequentially$\beta ^1$-continuous functionals is studied. We also show that $(L^1(K), \beta ^1)$ with the convolution multiplication is always a complete semitopological algebra, whereas it is a topological algebra if and only if $K$ is compact.

In this paper we introduce a vector-valued uncentred maximal operator in the setting of one-dimensional Bessel–Kingman hypergoups, and prove a maximal theorem for it.

The concept of semi-bounded generalized hypergroups (SBG hypergroups) is developed. These hypergroups are more special than generalized hypergroups introduced by Obata and Wildberger and more general than discrete hypergroups or even discrete signed hypergroups. The convolution of measures and functions is studied. In the case of commutativity we define the dual objects and prove some basic theorems of Fourier analysis. Furthermore, we investigate the relationship between orthogonal polynomials and generalized hypergroups. We discuss the Jacobi polynomials as an example.

In this article we study the Fourier space of a general hypergroup and its multipliers. The main result of this paper characterizes commutative hypergroups whose Fourier space forms a Banach algebra under pointwise product with an equivalent norm. Among those hypergroups whose Fourier space forms a Banach algebra, we identify a subclass for which the Gelfand spectrum of the Fourier algebra is equal to the underlying hypergroup. This subclass includes for instance, Jacobi hypergroups, Bessel-Kingman hypergroups.

In this paper we investigate when negative definite functions on commutative hypergroups satisfy the Schoenberg criterion.

A harmonic NA group is a suitable solvable extension of a two-step nilpotent Lie group N of Heisenberg type by R+, which acts on N by anisotropic dilations. A hypergroup is a locally compact space for which the space of Borel measures has a convolution structure preserving the probability measures and satisfying suitable conditions. We describe a class of hypergroups associated to NA groups.

In this paper we consider pseudo differential operators on local Hardy spaces hp (0 < p ≤ 1) on Chébli-Trimèche hypergroups of exponential growth.

In this paper we make use of semigroup methods on the space of compactly supported measures to obtain a Bochner representation for α-bounded positive-definite functions on a commutative hypergroup.