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

Let T_ 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 \Pₙ\₍ ₀ and \Rₙ\₍ ₀ (resp. ) satisfy R₍ (x) =T_{P₍+₁ (x) }₍+₁-₍-₁T_{ P₍-₁ (x) }₍-₁, n 2, where \ₙ\₍₁ is a sequence of non-zero complex numbers and ₂₍=2n, ₂₍-₁= 2n-1+ 2, n 1. In this contribution we focus the attention on the sequence \Sₙ^{ (, ) \}₍ ₀ of monic orthogonal polynomials with respect to the Dunkl-Sobolev inner product ₒ, =+, >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, ) =\ f; \ \|f\|ₔ^{2 + \| T f\|ₕ^2 <\}. Finally, two illustrative numerical examples are presented.