Laguerre expansions on conic domains

TL;DR

This paper develops Laguerre polynomial expansions on conic domains, providing explicit formulas for kernels, establishing a pseudo convolution structure, and analyzing convergence properties in $L^p$ spaces.

Contribution

It introduces a closed-form reproducing kernel and a bounded pseudo convolution structure for Laguerre expansions on conic surfaces, advancing harmonic analysis on such domains.

Findings

Explicit formula for the reproducing kernel

Bounded pseudo convolution structure in $L^p$ spaces

Convergence analysis of Cesàro means on conic domains

Abstract

We study the Fourier orthogonal expansions with respect to the Laguerre type weigh functions on the conic surface of revolution and the domain bounded by such a surface. The main results include a closed form formula for the reproducing kernels, which is the kernel of the orthogonal projection operator and a pseudo convolution structure on the conic domain; the latter is shown to be bounded in an appropriate $L^p$ space and used to study mean convergence of the Ces\`aro means of the Laguerre expansions on conic domains.