Weighted Sobolev orthogonal polynomials and approximation in the ball

2023/08/10 by Leonardo E. Figueroa, Figueroa, Leonardo E.
Engineering · Mathematics · Physics and Astronomy · #41A10 #41A25 #42C10 #46E35 #Classical Analysis and ODEs (math.CA) #Electromagnetic Scattering and Analysis #FOS: Mathematics #Mathematical functions and polynomials #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.2308.05469

openalex publication_date 2023/08/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish simultaneous approximation properties of weighted first-order Sobolev orthogonal projectors onto spaces of polynomials of bounded total degree in the Euclidean unit ball. The simultaneity is in the sense that we provide bounds for the projection error in both a weighted L² norm and a weighted H¹ seminorm, both involving the same weight of the generalized Gegenbauer type x ↦ (1-‖ x ‖²)^α, α> -1. The Sobolev orthogonal projectors producing the approximations are with respect to an alternative yet equivalent inner product for the corresponding uniformly weighted H¹ space. In order to obtain our approximation bounds, we study the orthogonal polynomial structure of this alternative Sobolev inner product obtaining, among other results, a characterization of its orthogonal polynomials as solutions of certain Sturm-Liouville problems. We do not rely on any particular basis of orthogonal polynomials, which allows for a streamlined and dimension-independent exposition.

Related

• Weighted Sobolev orthogonal polynomials on the unit ball
• Sobolev orthogonal polynomials defined via gradient on the unit ball
• Polynomial Approximation in Sobolev Spaces on the Unit Sphere and the Unit Ball
• Approximation and orthogonality in Sobolev spaces on a triangle
• A family of Sobolev Orthogonal Polynomials on the Unit Ball