Dual spaces vs. Haar measures of polynomial hypergroups

TL;DR

This paper investigates the relationship between dual spaces and Haar measures in polynomial hypergroups, establishing criteria for when the Haar measure weights are at least 2, and constructing examples where this fails.

Contribution

It provides sufficient conditions linking the dual space structure to Haar measure weights and introduces the first known examples with weights less than 2.

Findings

Many hypergroups have weights h(n) ≥ 2

Dual space interval condition implies h(n) ≥ 2

Constructed examples with h(1) < 2

Abstract

Many symmetric orthogonal polynomials $(P_n(x))_{n\in\mathbb{N}_0}$ induce a hypergroup structure on $\mathbb{N}_0$. The Haar measure is the counting measure weighted with $h(n):=1/\int_\mathbb{R}\!P_n^2(x)\,\mathrm{d}\mu(x)\geq1$, where $\mu$ denotes the orthogonalization measure. We observed that many naturally occurring examples satisfy the remarkable property $h(n)\geq2\;(n\in\mathbb{N})$. We give sufficient criteria and particularly show that $h(n)\geq2\;(n\in\mathbb{N})$ if the (Hermitian) dual space $\widehat{\mathbb{N}_0}$ equals the full interval $[-1,1]$, which is fulfilled by an abundance of examples. We also study the role of nonnegative linearization of products (and of the harmonic and functional analysis resulting from such expansions). Moreover, we construct two example types with $h(1)<2$. To our knowledge, these are the first such examples. The first type is based on Karlin-McGregor polynomials, and $\widehat{\mathbb{N}_0}$ consists of two intervals and can be chosen "maximal" in some sense; $h$ is of quadratic growth. The second type relies on certain compact operators; $h$ grows exponentially, and $\widehat{\mathbb{N}_0}$ is discrete.