Sobolev orthogonal polynomials on the unit ball via outward normal derivatives

TL;DR

This paper studies Sobolev orthogonal polynomials on the unit ball, focusing on their properties, explicit formulas, and asymptotic behavior, with applications to multivariate polynomial analysis.

Contribution

It introduces a new family of Sobolev orthogonal polynomials on the unit ball and derives explicit connection formulas, norm representations, and asymptotic properties.

Findings

Explicit connection formulas between classical and Sobolev polynomials

Representation formulas for norms and kernels

Asymptotic analysis of Christoffel functions

Abstract

We analyse a family of mutually orthogonal polynomials on the unit ball with respect to an inner product which involves the outward normal derivatives on the sphere. Using their representation in terms of spherical harmonics, algebraic and analytic properties will be deduced. First, we deduce explicit connection formulas relating classical multivariate ball polynomials and our family of Sobolev orthogonal polynomials. Then explicit representations for the norms and the kernels will be obtained. Finally, the asymptotic behaviour of the corresponding Christoffel functions is studied.