Sobolev orthogonal polynomials on product domains

TL;DR

This paper constructs Sobolev orthogonal polynomials on product domains with a specific inner product involving gradients and point evaluation, providing explicit bases for certain classical weight functions like Laguerre and Gegenbauer.

Contribution

It introduces a method to construct Sobolev orthogonal polynomials on product domains with explicit bases for specific weight functions, expanding the theory of such polynomials.

Findings

Explicit orthogonal bases for Sobolev polynomials with Laguerre weights

Explicit orthogonal bases for Sobolev polynomials with Gegenbauer weights

Method applicable to other weight functions and domains

Abstract

Orthogonal polynomials on the product domain $[a_1,b_1] \times [a_2,b_2]$ with respect to the inner product $$ \langle f,g \rangle_S = \int_{a_1}^{b_1} \int_{a_2}^{b_2} \nabla f(x,y)\cdot \nabla g(x,y)\, w_1(x)w_2(y) \,dx\, dy + \lambda f(c_1,c_2)g(c_1,c_2) $$ are constructed, where $w_i$ is a weight function on $[a_i,b_i]$ for $i = 1, 2$, $\lambda > 0$, and $(c_1, c_2)$ is a fixed point. The main result shows how an orthogonal basis for such an inner product can be constructed for certain weight functions, in particular, for product Laguerre and product Gegenbauer weight functions, which serve as primary examples.