Hyperinterpolation on the Sphere at the Minimal Projection Order

Abstract

dedicated to professor dr. dr. h. c. karl zeller on the occasion of his 75th birthdayWe investigate hyperinterpolation operators based on positive weighted quadrature rules, as introduced by Ian H. Sloan. If the rules are exact of double degree then, independently of the number of their nodes, the operator norms increase at the order of the minimal projections. Academic Press

Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds

“…In hyperinterpolation, functions are oversampled to avoid many of the inherent issues associated with trying to design an optimal collection of nodes for Lagrange interpolation [25,29,30,36]. This allows for functions to be approximated through L 2 -orthogonal projections using exact quadratures up to a desired order [2,30].…”

“…Hyperinterpolation methods use an oversampling of functions to grapple with some of the inherent challenges in designing optimal nodes for interpolation on general domains [25,29,30,36]. This allows for the treatment of approximation instead using approaches such as L 2 -orthogonal projection based on exact quadratures up to a desired order [2,30].…”

On generalized hyperinterpolation on the sphere

“…The following remarkable result was proved by [SW] (d = 3, under the regularity assumption (1.4)), [LS] (d arbitrary, under the regularity assumption (1.4)), and by [Re1] (d arbitrary, no additional regularity assumption):…”

“…We refer to [SW], [LS], [Re1] and [Re2] for the background information of hyperinterpolation on the sphere. In spite of the best-order result (1.5), pointwise convergence (for arbitrary f ∈ C(S d−1 )) cannot be attained by hyperinterpolation.…”

Conjectured inequalities for Jacobi polynomials and their largest zeros

“…Introduction. Special Jacobi polynomials P (α,β) n (x) with parameters β = α − 1 or β = α are frequently encountered in multivariate polynomial approximation on spherical surfaces, in which case α is related to the space dimension; see, e.g., [4], [5, §14.1]. Technical properties, especially inequalities, for these polynomials can be a valuable aid in simplifying various estimates in the theory of spherical approximation.…”