Localized linear polynomial operators and quadrature formulas on the sphere

TL;DR

This paper develops adaptive, localized polynomial operators on the sphere using scattered data, with new quadrature formulas that are exact for high-degree spherical polynomials, outperforming traditional methods.

Contribution

It introduces universal, auto-adaptive operators and high-degree quadrature formulas based on scattered data, enhancing approximation on the sphere.

Findings

Operators show superior approximation and localization properties.

Quadrature formulas are exact for spherical polynomials up to degree 178.

Numerical experiments confirm the effectiveness over traditional methods.

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.