Weighted Sobolev orthogonal polynomials on the unit ball

TL;DR

This paper constructs a new family of mutually orthogonal polynomials on the unit ball with respect to a weighted Sobolev inner product, using spherical harmonics and recursive formulas, extending classical polynomial theory.

Contribution

It introduces Sobolev orthogonal polynomials on the unit ball with a weighted inner product involving gradients, providing explicit recursive formulas for their generation.

Findings

Orthogonal polynomials are explicitly constructed using spherical harmonics.

Recursive formulas enable efficient generation of these polynomials.

The approach extends classical orthogonal polynomial theory to Sobolev spaces.

Abstract

For the weight function $W_\mu(x) = (1-|x|^2)^\mu$, $\mu > -1$, $\lambda > 0$ and $b_\mu$ a normalizing constant, a family of mutually orthogonal polynomials on the unit ball with respect to the inner product $$ \la f,g \ra = {b_\mu [\int_{\BB^d} f(x) g(x) W_\mu(x) dx + \lambda \int_{\BB^d} \nabla f(x) \cdot \nabla g(x) W_\mu(x) dx]} $$ are constructed in terms of spherical harmonics and a sequence of Sobolev orthog onal polynomials of one variable. The latter ones, hence, the orthogonal polynomials with respect to $\la \cdot,\cdot\ra$, can be generated through a recursive formula.