Optimal Lebesgue constants for least squares polynomial approximation on the (hyper)sphere

2018/08/10 by Woula Themistoclakis, Themistoclakis, Woula, Marc Van Barel +1
Computer Science · Mathematics · #33-C45 #41-A10 #65-D99 #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical functions and polynomials #Matrix Theory and Algorithms #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1808.03530

openalex publication_date 2018/08/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate the uniform approximation provided by least squares polynomials on the unit Euclidean sphere \mathbbS^q in ℝ^q+1, with q≥ 2. Like any other polynomial projection, the study concerns the growth, as the degree n tends to infinity, of the associated Lebesgue constant, i.e., of the uniform norm of the least squares operator. If the least squares polynomial of degree n is based on a set of points, which are nodes of a positive weighted quadrature rule of degree of exactness 2n, then we state two different sufficient conditions for having an optimal Lebesgue constant that increases with n at the minimal projections order. Hence, under our assumptions least squares and hyperinterpolation polynomials provide a comparable approximation with respect to the uniform norm.

Related

• Uniform approximation on the sphere by least squares polynomials
• Approximation and quadrature by weighted least squares polynomials on the sphere
• Weighted ℓ_q approximation problems on the ball and on the sphere
• Evaluating Lebesgue constants by Chebyshev polynomial meshes on cube, simplex and ball
• Minimal projections onto spaces of polynomials on real euclidean spheres