Summability of Fourier orthogonal series for Jacobi weight functions on the simplex in <IMG ALIGN=BOTTOM HEIGHT=17 WIDTH=22>期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Summability of Fourier orthogonal series for Jacobi weight functions on the simplex in

Authors:	Yuan Xu

Institution:	Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222

Abstract:	We study the Fourier expansion of a function in orthogonal polynomial series with respect to the weight functions $\begin{displaymath}x_{1}^{\alpha _{1} -1/2} \cdots x_{d}^{\alpha _{d} -1/2}(1-\|\mathbf{x}\|_{1})^{\alpha _{d+1}-1/2}\end{displaymath}$ on the standard simplex $\Sigma ^{d}$ in $\mathbb{R}^{d}$ . It is proved that such an expansion is uniformly $(C, \delta )$ summable on the simplex for any continuous function if and only if $\delta > \|\alpha \|_{1} + (d-1)/2$ . Moreover, it is shown that $(C, \|\alpha \|_{1} + (d+1)/2)$ means define a positive linear polynomial identity, and the index is sharp in the sense that $(C,\delta )$ means are not positive for $0 <\delta <\|\alpha \|_{1} + (d+1)/2$ .

Keywords:	Orthogonal polynomials in several variables on simplex Ces\`{a}ro summability positive kernel

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