Dunkl symmetric coherent pairs of measures. An application to Fourier Dunkl-Sobolev expansions

TL;DR

This paper studies symmetric Dunkl-coherent pairs of measures and their associated orthogonal polynomials, introducing Fourier-Dunkl-Sobolev expansions and providing algorithms and examples for practical computation.

Contribution

It introduces a new class of Dunkl-coherent pairs and develops Fourier-Dunkl-Sobolev expansions with algorithms for their coefficients, along with numerical illustrations.

Findings

Defined symmetric Dunkl-coherent pairs of measures.

Developed algorithms for Fourier-Dunkl-Sobolev expansions.

Provided numerical examples demonstrating the methods.

Abstract

Let $\mathcal{T}_{\mu}$ be the Dunkl operator. A pair of symmetric measures $(u, v)$ supported on a symmetric subset of the real line is said to be a symmetric Dunkl-coherent pair if the corresponding sequences of monic orthogonal polynomials $\{P_n\}_{n\geq 0}$ and $\{R_n\}_{n\geq 0}$ (resp.) satisfy $$ R_{n}(x)=\frac{\mathcal{T}_{\mu}P_{n+1} (x)}{\mu_{n+1}}-\sigma_{n-1}\frac{\mathcal{T}_{\mu} P_{n-1}(x)}{\mu_{n-1}}, n\geq 2,$$ where $\{\sigma_n\}_{n\geq1}$ is a sequence of non-zero complex numbers and $\mu_{2n}=2n, \mu_{2n-1}= 2n-1+ 2\mu, n\geq 1.$ In this contribution we focus the attention on the sequence $\{S_n^{(\lambda,\mu)}\}_{n\geq 0}$ of monic orthogonal polynomials with respect to the Dunkl-Sobolev inner product $$ <p,q>_{s,\mu}=<u,pq>+\lambda<v,\mathcal{T}_{\mu}p\mathcal{T}_{\mu}q>, \lambda >0, \ \ p, \ q \ \in \mathcal{P}.$$ An algorithm is stated to compute the coefficients of the Fourier--Sobolev type expansions with respect to $<. , .>$ for suitable smooth functions $f$ such that $f \in\mathcal{W}_2^1(R, u, v, \mu)=\{ f; \ \|f\|_{u}^{2} + \lambda \| \mathcal{T}_{\mu }f\|_{v}^{2} <\infty\}$. Finally, two illustrative numerical examples are presented.