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

TL;DR

This paper investigates the conditions under which Fourier expansions using Heckman-Opdam polynomials of type A1 converge in L^p spaces, providing specific p-range bounds for convergence.

Contribution

It establishes the precise p-range for L^p-convergence of Fourier-Heckman-Opdam expansions of type A1 using kernel estimates and duality.

Findings

Partial sums converge in L^p for 2 - 1/(k+1) < p < 2 + 1/k

Provides kernel estimates and duality arguments for convergence

Defines explicit p-range bounds for convergence

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([-\pi,\pi],dm_k)$ for $$2-\frac{1}{k+1} < p < 2+\frac{1}{k}.$$