Harmonic analysis of little $q$-Legendre and associated symmetric Pollaczek polynomials

Stefan Kahler

Published 2018-06-01Version 1

An elegant and fruitful way to bring harmonic analysis into the theory of orthogonal polynomials and special functions---or, from the opposite point of view, to associate certain Banach algebras with orthogonal polynomials satisfying a specific (but frequently satisfied) nonnegative linearization property---is the concept of a polynomial hypergroup. Polynomial hypergroups (or the underlying polynomials, respectively) are accompanied by $L^1$-algebras and a rich, well-developed and unified harmonic analysis. However, the individual behavior strongly depends on the underlying polynomials. We study two classes which are very different to each other, in particular with regard to amenability properties of the corresponding $L^1$-algebras: concerning the little $q$-Legendre polynomials, which are orthogonal with respect to a purely discrete measure and whose $L^1$-algebras have been known to be right character amenable, we will show that the $L^1$-algebras are spanned by their idempotents and hence also weakly amenable. Concerning the associated symmetric Pollaczek polynomials, which are a two-parameter generalization of the ultraspherical polynomials and come with an absolutely continuous measure, we will provide complete characterizations of right character amenability, weak amenability and point amenability (i.e., the global nonexistence of nonzero bounded point derivations), and we shall see that there is a large parameter region for which none of these amenability properties holds. While the crucial underlying nonnegative linearization property has been known to be satisfied for the little $q$-Legendre polynomials, the analogue problem for the associated symmetric Pollaczek polynomials will be solved in this paper. Our strategy relies on chain sequences, continued fractions, Tur\'{a}n type inequalities, character estimations, suitable transformations and asymptotic behavior.