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arXiv · 1806.00339

Tur\'{a}n's inequality, nonnegative linearization and amenability properties for associated symmetric Pollaczek polynomials

Stefan Kahler
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Abstract

An elegant and fruitful way to bring harmonic analysis into the theory of orthogonal polynomials and special functions, or 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 the associated symmetric Pollaczek polynomials, which are a two-parameter generalization of the ultraspherical polynomials. Considering the associated $L^1$-algebras, we will provide complete characterizations of weak amenability and point amenability by specifying the corresponding parameter regions. In particular, we shall see that there is a large parameter region for which none of these amenability properties holds (which is very different to $L^1$-algebras of locally compact groups). Moreover, we will rule out right character amenability. The crucial underlying nonnegative linearization property will be established, too, which particularly establishes a conjecture of R. Lasser (1994). Furthermore, we shall prove Tur\'{a}n's inequality for associated symmetric Pollaczek polynomials. Our strategy relies on chain sequences, asymptotic behavior, further Tur\'{a}n type inequalities and transformations into more convenient orthogonal polynomial systems.

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BibTeXRIS

Stefan Kahler. 2018-06-01. Tur\'{a}n's inequality, nonnegative linearization and amenability properties for associated symmetric Pollaczek polynomials. https://doi.org/10.1016/j.jat.2021.105580

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