Ces\`aro means of orthogonal expansions in several variables

TL;DR

This paper investigates Cesàro means for orthogonal expansions across multiple variables on spheres, balls, and simplices, providing sharp estimates and exact norms for these means and projection operators.

Contribution

It introduces precise pointwise estimates for Cesàro kernels and projection operators in multivariable orthogonal expansions, advancing understanding of their behavior on various domains.

Findings

Established sharp pointwise estimates for Cesàro kernels.

Derived exact order of norms for Cesàro means and projections.

Extended results to multiple domains including spheres, balls, and simplices.

Abstract

Ces\`aro $(C,\delta)$ means are studied for orthogonal expansions with respect to the weight function $\prod_{i=1}^{d}|x_i|^{2\k_i}$ on the unit sphere, and for the corresponding weight functions on the unit ball and the Jacobi weight on the simplex. A sharp pointwise estimate is established for the $(C,\d)$ kernel with $\d > -1$ and for the kernel of the projection operator, which allows us to derive the exact order for the norm of the Ces\`aro means and the projection operator on these domains.