Ces\`aro means of Jacobi expansions on the parabolic biangle

TL;DR

This paper investigates the boundedness and positivity of Cesàro means for two-variable Jacobi polynomial expansions on a specific geometric domain called the parabolic biangle, extending understanding of their convergence properties.

Contribution

The paper establishes conditions under which Cesàro means are uniformly bounded and positive for Jacobi expansions on the parabolic biangle, using a convolution interpretation based on a product formula.

Findings

Cesàro means are uniformly bounded if elta>lpha+eta+1

Cesàro means are positive linear operators if eltalpha+2eta+3

The study extends classical results to a two-variable setting on a non-standard domain.

Abstract

We study Ces\`aro $(C,\delta)$ means for two-variable Jacobi polynomials on the parabolic biangle $B=\{(x_1,x_2)\in{\mathbb R}^2:0\leq x_1^2\leq x_2\leq 1\}$. Using the product formula derived by Koornwinder & Schwartz for this polynomial system, the Ces\`aro operator can be interpreted as a convolution operator. We then show that the Ces\`aro $(C,\delta)$ means of the orthogonal expansion on the biangle are uniformly bounded if $\delta>\alpha+\beta+1$, $\alpha-\frac 12\geq\beta\geq 0$. Furthermore, for $\delta\geq\alpha+2\beta+\frac 32$ the means define positive linear operators.