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Harmonic analysis of little $q$-Legendre polynomials

Stefan Kahler

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.

Harmonic analysis of little $q$-Legendre polynomials

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 -algebras, associated with underlying orthogonal polynomials. The individual behavior strongly depends on these underlying polynomials. We study the little -Legendre polynomials, which are orthogonal with respect to a discrete measure. We will show that their -algebras have the property that every element can be approximated by linear combinations of idempotents. This particularly implies that these -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 -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 -Legendre polynomials yield the first example of a polynomial hypergroup whose -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.

Paper Structure

This paper contains 7 sections, 14 theorems, 87 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Underlying setting and basics about polynomial hypergroups and their Fourier analysis
  4. Little q-Legendre polynomials and outline of the paper
  5. Character estimations, continued fractions and uniform boundedness properties
  6. Density of idempotents
  7. Amenability properties

Key Result

Theorem \oldthetheorem

Let $q\in(0,1)$ and $P_n(x)=R_n(x;q)\;(n\in\mathbb{N}_0)$. Then there is some $C>0$ such that for all $n\in\mathbb{N}_0$. It is possible to take

Figures (3)

  • Figure 1: Left: $\left\|\alpha_{1-q^n}\right\|_1/\left\|\alpha_{1-q^n}\right\|_2^2$ for $n\in\{0,\ldots,19\}$ and $q=2/3$. Right: $\left\|\alpha_x\right\|_1/\left\|\alpha_x\right\|_2^2$ for $x\in\{1-q^n:n\in\{0,\ldots,19\}\}$ and $q=2/3$. The explicit bound provided by Theorem \ref{['thm:characternorm']} is $\approx20$.
  • Figure 2: Left (visualization of \ref{['eq:ratioasymptoticsuni']}): $\lvert\alpha_{1-q^n}(n+k+1)/(\alpha_{1-q^n}(n+k)q^{k+1})\rvert$ for $k\in\{0,\ldots,7\}$, $n\in\{0,\ldots,9\}$ and $q=2/3$ (so $K=3$). Right (visualization of \ref{['eq:asymptoticsuni']}): $\lvert\alpha_{1-q^n}(n+k)\rvert$ for $k\in\{0,\ldots,7\}$, $n\in\{0,\ldots,9\}$ and $q=2/3$ (so $K=3$); the curve corresponds to the function $x\mapsto4^{x-K}q^{(K+x+1)(x-K)/2}$.
  • Figure 3: $\left\|\epsilon_1-\left(\epsilon_0-\sum_{n=0}^N\widehat{\epsilon_0-\epsilon_1}(1-q^n)\frac{\alpha_{1-q^n}}{\left\|\alpha_{1-q^n}\right\|_2^2}\right)\right\|_1$ for $N\in\{0,\ldots,9\}$ and $q=2/3$.

Theorems & Definitions (23)

  • Theorem \oldthetheorem
  • Proposition 2.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • proof : Proof (Lemma \ref{['lma:characterdecay']})
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 13 more