Title: Dunkl symmetric coherent pairs of measures. An application to Fourier Dunkl-Sobolev expansions Show Chinese title

Title: Dunkl对称相干对的测度。 Dunkl-Sobolev展开在傅里叶中的应用

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. Show Chinese abstract

Abstract: 设$\mathcal{T}_{\mu}$为Dunkl算子。 一对在实数线的对称子集上支撑的对称测度$(u, v)$被称为对称Dunkl相容对，如果对应的首一正交多项式序列$\{P_n\}_{n\geq 0}$和$\{R_n\}_{n\geq 0}$（分别） 满足$$ R_{n}(x)=\frac{\mathcal{T}_{\mu}P_{n+1} (x)}{\mu_{n+1}}-\sigma_{n-1}\frac{\mathcal{T}_{\mu} P_{n-1}(x)}{\mu_{n-1}}, n\geq 2,$$ 其中$\{\sigma_n\}_{n\geq1}$是一组非零复数序列，且 $\mu_{2n}=2n, \mu_{2n-1}= 2n-1+ 2\mu, n\geq 1.$ 在本次贡献中，我们关注序列 $\{S_n^{(\lambda,\mu)}\}_{n\geq 0}$的首一正交多项式，相对于 Dunkl-Sobolev 内积$$ <p,q>_{s,\mu}=<u,pq>+\lambda<v,\mathcal{T}_{\mu}p\mathcal{T}_{\mu}q>, \lambda >0, \ \ p, \ q \ \in \mathcal{P}.$$ 给出了一个算法，用于计算相对于$<. , .>$的傅里叶-索博列夫型展开的系数，对于合适的光滑函数$f$使得$f \in\mathcal{W}_2^1(R, u, v, \mu)=\{ f; \ \|f\|_{u}^{2} + \lambda \| \mathcal{T}_{\mu }f\|_{v}^{2} <\infty\}$。 最后，给出两个示例数值例子。