Orthogonal structure and orthogonal series in and on a double cone or a hyperboloid

TL;DR

This paper studies orthogonal polynomials on double cones and hyperboloids, revealing their eigenfunction properties, addition formulas, and convolution structures, and explores their relation to Hermite polynomials.

Contribution

It introduces new families of orthogonal polynomials on these surfaces, characterizes their properties, and establishes their connections to classical polynomials like Hermite.

Findings

Orthogonal polynomials are eigenfunctions of second order differential operators.

Addition formulas provide closed-form reproducing kernels.

Convolution structures facilitate Fourier series analysis on these domains.

Abstract

We consider orthogonal polynomials on the surface of a double cone or a hyperboloid of revolution, either finite or infinite in axis direction, and on the solid domain bounded by such a surface and, when the surface is finite, by hyperplanes at the two ends. On each domain a family of orthogonal polynomials, related to the Gegebauer polynomials, is study and shown to share two characteristic properties of spherical harmonics: they are eigenfunctions of a second order linear differential operator with eigenvalues depending only on the polynomial degree, and they satisfy an addition formula that provides a closed form formula for the reproducing kernel of the orthogonal projection operator. The addition formula leads to a convolution structure, which provides a powerful tool for studying the Fourier orthogonal series on these domains. Furthermore, another family of orthogonal polynomials, related to the Hermite polynomials, is defined and shown to be the limit of the first family, and their properties are derived accordingly.