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

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 $\{P_n{\}}_{n\ge 0}$ and $\{R_n{\}}_{n\ge 0}$ (resp.) satisfy $$R_n(x)=\frac{{\mathcal{T}}_{\mu }{P}_{n+1}(x)}{{\mu }_{n+1}}-{\sigma }_{n-1}\frac{{\mathcal{T}}_{\mu }{P}_{n-1}(x)}{{\mu }_{n-1}},n\ge 2,$$ where $\{{\sigma }_n{\}}_{n\ge 1}$ is a sequence of non-zero complex numbers and ${\mu }_{2n}=2n,{\mu }_{2n-1}=2n-1+2\mu ,n\ge 1.$ In this contribution we focus the attention on the sequence $\{S_n^{(\lambda ,\mu )}{\}}_{n\ge 0}$ 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 $f\in {\mathcal{W}}_2^1(R,u,v,\mu )=\{f;\Vert f{\Vert }_u^2+\lambda \Vert {\mathcal{T}}_{\mu }f{\Vert }_v^2<\infty \}$. Finally, two illustrative numerical examples are presented.