Cesàro means of Jacobi expansions on the parabolic biangle

Abstract

We study Cesàro $(C,\delta )$ means for two-variable Jacobi polynomials on the parabolic biangle $B=\{(x_1,x_2)\in {\mathbb{R}}^2:0\le x_1^2\le x_2\le 1\}$. Using the product formula derived by Koornwinder & Schwartz for this polynomial system, the Cesàro operator can be interpreted as a convolution operator. We then show that the Cesàro $(C,\delta )$ means of the orthogonal expansion on the biangle are uniformly bounded if $\delta >\alpha +\beta +1$, $\alpha -\frac{1}{2}\ge \beta \ge 0$. Furthermore, for $\delta \ge \alpha +2\beta +\frac{3}{2}$ the means define positive linear operators.