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

Let Tμ 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 ₙ≥ 0 and ₙ≥ 0 (resp.) satisfy Rₙ(x)=TμPₙ₊₁ (x)μₙ₊₁-σₙ₋₁Tμ Pₙ₋₁(x)μₙ₋₁, n≥ 2, where \σₙ≥1 is a sequence of non-zero complex numbers and μ₂ₙ=2n, μ₂ₙ₋₁= 2n-1+ 2μ, n≥ 1. In this contribution we focus the attention on the sequence ₙ(λ,μ)≥ 0 of monic orthogonal polynomials with respect to the Dunkl-Sobolev inner product <p,q>s,μ=<u,pq>+λ<v,TμpTμq>, λ >0, \ \ p, \ q \ ∈ 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 ₂¹(R, u, v, μ)=\ f; \ \|f\|ᵤ² + λ \| Tμf\|ᵥ² <∞\. Finally, two illustrative numerical examples are presented.