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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2405.14771 |
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| _version_ | 1866916257838661632 |
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| author | Sghaier, Mabrouk Marcellán, Francisco |
| author_facet | Sghaier, Mabrouk Marcellán, Francisco |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_14771 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dunkl symmetric coherent pairs of measures. An application to Fourier Dunkl-Sobolev expansions Sghaier, Mabrouk Marcellán, Francisco Classical Analysis and ODEs Primary 42C05, Secondary 33C45, 42C10 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. |
| title | Dunkl symmetric coherent pairs of measures. An application to Fourier Dunkl-Sobolev expansions |
| topic | Classical Analysis and ODEs Primary 42C05, Secondary 33C45, 42C10 |
| url | https://arxiv.org/abs/2405.14771 |