Localized linear polynomial operators and quadrature formulas on the sphere

Abstract

The purpose of this paper is to construct universal, auto--adaptive, localized, linear, polynomial (-valued) operators based on scattered data on the (hyper--)sphere $\SS^q$ ($q\ge 2$). The approximation and localization properties of our operators are studied theoretically in deterministic as well as probabilistic settings. Numerical experiments are presented to demonstrate their superiority over traditional least squares and discrete Fourier projection polynomial approximations. An essential ingredient in our construction is the construction of quadrature formulas based on scattered data, exact for integrating spherical polynomials of (moderately) high degree. Our formulas are based on scattered sites; i.e., in contrast to such well known formulas as Driscoll--Healy formulas, we need not choose the location of the sites in any particular manner. While the previous attempts to construct such formulas have yielded formulas exact for spherical polynomials of degree at most 18, we are able to construct formulas exact for spherical polynomials of degree 178.

24 pages 2 figures, accepted for publication in SIAM J. Numer. Anal

Cited by in corpus (14)

• Numerical methods for computing Casimir interactions
• Signal reconstruction from the magnitude of subspace components
• Tight framelets and fast framelet filter bank transforms on manifolds
• Quasi Monte Carlo integration and kernel-based function approximation on Grassmannians
• Tikhonov regularization for polynomial approximation problems in Gauss quadrature points
• Minimum Sobolev norm interpolation of derivative data
• Fast Cross-validation in Harmonic Approximation
• FaVeST: Fast Vector Spherical Harmonic Transforms
• Function approximation by deep networks
• Eignets for function approximation on manifolds
• Nonparametric regression using needlet kernels for spherical data
• Function approximation with zonal function networks with activation functions analogous to the rectified linear unit functions
• Optimal Lebesgue constants for least squares polynomial approximation on the (hyper)sphere
• Lasso hyperinterpolation over general regions