Dual spaces vs. Haar measures of polynomial hypergroups

Journal of Approximation Theory, vol. 305, pp. 106099

Abstract

Many symmetric orthogonal polynomials ( P n ( x ) ) n ∈ N 0 induce a hypergroup structure on N 0 . The Haar measure is the counting measure weighted with h ( n ) ≔ 1 / ∫ R P n 2 ( x ) d μ ( 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 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 resulting harmonic and functional analysis). 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 N 0 ̂ consists of two intervals and can be chosen “maximal” in some sense; h is of quadratic growth. The second type relies on hypergroups of strong compact type; h grows exponentially, and N 0 ̂ is discrete.

Authors 2

1. Stefan Kahler corresponding Aachen Fachgruppe Mathematik Lehrstuhl A für Mathematik

   RWTH Aachen University

   Affiliation as printed

   Fachgruppe Mathematik, RWTH Aachen University, Pontdriesch 14-16, 52062 Aachen, Germany

   Lehrstuhl A für Mathematik, RWTH Aachen University, 52056 Aachen, Germany

2. Ryszard Szwarc

   University of Wrocław

   Affiliation as printed

   Institute of Mathematics, Wrocław University, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland