Fourier orthogonal series on a paraboloid

Abstract

We study orthogonal structures and Fourier orthogonal series on the surface of a paraboloid $\mathbb{V}_0^{d+1} = \{(x,t): \|x\| = \sqrt{t}, \, x \in \mathbb{R}^d, \, 0\le t<1\}$. The reproducing kernels of the orthogonal polynomials with respect to $t^\beta(1-t)^\gamma$ on $\mathbb{V}_0^{d+1}$ are related to the reproducing kernels of the Jacobi polynomials on the parabolic domain $\{(x_1,x_2): x_1^2 \le x_2 \le 1\}$ in $\mathbb{R}^2$. This connection serves as an essential tool for our study of the Fourier orthogonal series on the surface of the paraboloid, which allow us, in particular, to study the convergence of the Ces\`aro means on the surface. Analogous results are also established for the solid paraboloid bounded by $\mathbb{V}_0^{d+1}$ and the hyperplane $t=1$.