Fourier orthogonal series on a paraboloid

TL;DR

This paper develops orthogonal polynomial and Fourier series theory on paraboloid surfaces, establishing kernel relations and convergence results for Cesàro means, extending to solid paraboloids.

Contribution

It introduces a novel connection between orthogonal kernels on paraboloids and Jacobi polynomials, enabling analysis of Fourier series convergence on these surfaces.

Findings

Reproducing kernels relate to Jacobi polynomial kernels.

Convergence of Cesàro means on paraboloid surfaces is established.

Results extend to solid paraboloids bounded by the surface.

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 $t=1$.