$L^p$-Convergence of Fourier-Heckman-Opdam Expansions

TL;DR

This paper addresses the L^p convergence of Fourier expansions built from non-symmetric Heckman-Opdam polynomials of type $A_1$. It develops a kernel-based approach and duality arguments to prove convergence of the partial sums $S_N(f)$ for functions in $L^p([-\,\pi,\pi],dm_k)$ within the sharp range $2-\frac{1}{k+1}<p<2+\frac{1}{k}$, and it furnishes a counterexample outside this interval. The results rely on explicit formulas and norms for $E_n^k$, the relation to $P_n^k$, and careful kernel estimates involving ultraspherical polynomials. This extends harmonic analysis on root-system-related Fourier expansions and clarifies convergence behavior for these generalized orthogonal bases.

Abstract

We study the $L^p$-convergence of Fourier expansions in terms of non-symmetric Heckman-Opdam polynomials of type $A_1$. Using kernel estimates and duality arguments, we prove that the partial sums converge in $ L^p([-π,π],dm_k)$ for $$2-\frac{1}{k+1} < p < 2+\frac{1}{k}.$$

Theorems & Definitions (19)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Theorem 3.1: Main result
• Lemma 3.2
• proof
• Lemma 3.3