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

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.