Sobolev orthogonal polynomials on the conic surface

TL;DR

This paper extends the theory of orthogonal polynomials on conic surfaces to Sobolev inner products, providing explicit bases, projection formulas, and convergence analysis, especially for eigenfunctions of a second order differential operator.

Contribution

It introduces Sobolev orthogonal polynomials on conic surfaces, constructs explicit bases, and analyzes their convergence properties, extending previous work on classical orthogonal polynomials.

Findings

Explicit orthogonal basis constructed

Projection operators derived and analyzed

Sobolev orthogonal polynomials are eigenfunctions of a differential operator

Abstract

Orthogonal polynomials with respect to the weight function $w_{\beta,\gamma}(t) = t^\beta (1-t)^\gamma$, $\gamma > -1$, on the conic surface $\{(x,t): \|x\| = t, \, x \in \mathbb{R}^d, \, t \le 1\}$ are studied recently, and they are shown to be eigenfunctions of a second order differential operator $\mathcal{D}_\gamma$ when $\beta =-1$. We extend the setting to the Sobolev inner product, defined as the integration of the $s$-th partial derivatives in $t$ variable with respect to $w_{\beta+s,0}$ over the conic surface plus a sum of integrals over the rim of the cone. Our main results provide an explicit construction of an orthogonal basis and a formula for the orthogonal projection operators; the latter is used to exploit the interaction of differential operators and the projection operator, which allows us to study the convergence of the Fourier orthogonal series. The study can be regarded as an extension of the orthogonal structure to the weight function $w_{\beta, -s}$ for a positive integer $s$. It shows, in particular, that the Sobolev orthogonal polynomials are eigenfunctions of $\mathcal{D}_{\gamma}$ when $\gamma = -1$.