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 ${P_n}{n\geq 0}$ 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.