Hypergroups on the Set of All Integers

Stating a two-parameter class of examples, a — positive — answer is given to the question “Are there any nontrivial hypergroups on Z?” . 1. Background and Introduction In 1989 Zeuner [14] proved that no nontrivial hypergroups (i. e. not the group) exist on the set of real numbers R. Five years later Rösler [5] showed that by weakening the axioms by considering signed hypergroups some of these structures can be found on R. Hence, one of the next questions asked was whether any analogue results hold for the discrete case, the set of integers Z. The reason why the existence of nontrivial hypergroups on Z has stayed an open problem was that Zeuner’s argumentation for R is based on topological properties and cannot be transferred to Z. On the other hand, no examples were found. Rösler [7] examined signed hypergroups on Z induced by trigonometric polynomials orthogonal w. r. t. Jacobi weights and proved that no nontrivial hypergroups evolve for these weights. After a short course on discrete signed hypergroups we will give a more general setting for obtaining these structures on Z. They are induced by a system of trigonometric polynomials orthogonal on the unit circle T w. r. t. a symmetric probability measure in an analogue way as polynomial signed hypergroups on the set N0 := {n ∈ Z : n ≥ 0}. The special case of Bernstein-Szegö weights on T always fulfills this condition and if the corresponding parameters are in a certain range a nontrivial hypergroup on Z is obtained. 2. Signed Hypergroups Let K 6= ∅ be a discrete set. For an element x of K denote εx the Dirac function at x and as usually the space ` := {f = ∑∞ n=1 anεxn : an ∈ C, xn ∈ K, ∑∞ n=1 |an| < ∞} with total variation norm ‖f‖ := ∑∞ n=1 |an| for f ∈ `. Moreover, define for a fixed M ≥ 1 the space `M of real, possibly signed measures on K with finite support and total variation less than or equal to M such that the measure of K is one. Due to the last two conditions in the definition any f ∈ `1 has nonnegative weights an so that `1 is the set of probability measures on K with finite support. Date: December 3, 2002. 1991 Mathematics Subject Classification. Primary 20N20, 33B10.