Dual spaces vs. Haar measures of polynomial hypergroups

Many symmetric orthogonal polynomials (Pn(x))n∈N0 induce a hypergroup structure on N0. The Haar measure is the counting measure weighted with h(n)≔1/∫RPn2(x)dμ(x)≥1, where μ denotes the orthogonalization measure. We observed that many naturally occurring examples satisfy the remarkable property h(n)...

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Bibliographic Details
Published in:Journal of approximation theory 2025-01, Vol.305, p.106099, Article 106099
Main Authors: Kahler, Stefan, Szwarc, Ryszard
Format: Article
Language:English
Subjects:
Dual space
Haar measure
Hypergroups
Orthogonal polynomials
Polynomial estimates
Polynomial expansions
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Summary:Many symmetric orthogonal polynomials (Pn(x))n∈N0 induce a hypergroup structure on N0. The Haar measure is the counting measure weighted with h(n)≔1/∫RPn2(x)dμ(x)≥1, where μ denotes the orthogonalization measure. We observed that many naturally occurring examples satisfy the remarkable property h(n)≥2(n∈N). We give sufficient criteria and particularly show that h(n)≥2(n∈N) if the (Hermitian) dual space N0̂ 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 resulting harmonic and functional analysis). Moreover, we construct two example types with h(1)
ISSN:0021-9045
DOI:10.1016/j.jat.2024.106099