Sparad:
| Huvudupphovsmän: | , |
|---|---|
| Materialtyp: | Preprint |
| Publicerad: |
2024
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| Ämnen: | |
| Länkar: | https://arxiv.org/abs/2405.14771 |
| Taggar: |
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Innehållsförteckning:
- Let $\mathcal{T}_μ$ 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 $$ R_{n}(x)=\frac{\mathcal{T}_μP_{n+1} (x)}{μ_{n+1}}-σ_{n-1}\frac{\mathcal{T}_μ P_{n-1}(x)}{μ_{n-1}}, n\geq 2,$$ where $\{σ_n\}_{n\geq1}$ is a sequence of non-zero complex numbers and $μ_{2n}=2n, μ_{2n-1}= 2n-1+ 2μ, n\geq 1.$ In this contribution we focus the attention on the sequence $\{S_n^{(λ,μ)}\}_{n\geq 0}$ of monic orthogonal polynomials with respect to the Dunkl-Sobolev inner product $$ <p,q>_{s,μ}=<u,pq>+λ<v,\mathcal{T}_μp\mathcal{T}_μq>, λ>0, \ \ p, \ q \ \in \mathcal{P}.$$ An algorithm is stated to compute the coefficients of the Fourier--Sobolev type expansions with respect to $<. , .>$ for suitable smooth functions $f$ such that $f \in\mathcal{W}_2^1(R, u, v, μ)=\{ f; \ \|f\|_{u}^{2} + λ\| \mathcal{T}_{μ}f\|_{v}^{2} <\infty\}$. Finally, two illustrative numerical examples are presented.