Harmonic analysis of little $q$-Legendre polynomials

RWTH Publications (RWTH Aachen)

Abstract

Many classes of orthogonal polynomials satisfy a specific linearization property giving rise to a polynomial hypergroup structure, which offers an elegant and fruitful link to Fourier analysis, harmonic analysis and functional analysis. From the opposite point of view, this allows regarding certain Banach algebras as $L^1$-algebras, associated with underlying orthogonal polynomials. The individual behavior strongly depends on these underlying polynomials. We study the little $q$-Legendre polynomials, which are orthogonal with respect to a discrete measure. We will show that their $L^1$-algebras have the property that every element can be approximated by linear combinations of idempotents. This particularly implies that these $L^1$-algebras are weakly amenable (i. e., every bounded derivation into the dual module is an inner derivation), which is known to be shared by any $L^1$-algebra of a locally compact group; in the polynomial hypergroup context, weak amenability is rarely satisfied and of particular interest because it corresponds to a certain property of the derivatives of the underlying polynomials and their (Fourier) expansions w.r.t. the polynomial basis. To our knowledge, the little $q$-Legendre polynomials yield the first example of a polynomial hypergroup whose $L^1$-algebra is weakly amenable and right character amenable but fails to be amenable. As a crucial tool, we establish certain uniform boundedness properties of the characters. Our strategy relies on the Fourier transformation on hypergroups, the Plancherel isomorphism, continued fractions, character estimations and asymptotic behavior.

Authors 1

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

   RWTH Aachen University · Technical University of Munich

   Affiliation as printed

   . München, Germany

   Department of Mathematics, Chair for Mathematical Modelling, Technical University of Munich, Boltzmannstr. 3, 85747 Garching

   Fachgruppe Mathematik, RWTH Aachen University, Pontdriesch 14-16, 52062 Aachen, Ger-many

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