Bypassing the quadrature exactness assumption of hyperinterpolation on the sphere

TL;DR

This paper introduces unfettered hyperinterpolation on the sphere, removing the quadrature exactness requirement by using the Marcinkiewicz--Zygmund property, and provides error estimates supported by numerical experiments.

Contribution

It proposes a new hyperinterpolation scheme that relaxes quadrature exactness constraints using the Marcinkiewicz--Zygmund property, with proven error bounds.

Findings

Error estimates for unfettered hyperinterpolation are established.

Numerical experiments confirm the theoretical error bounds.

Refined error estimates are available for QMC design points.

Abstract

This paper focuses on the approximation of continuous functions on the unit sphere by spherical polynomials of degree $n$ via hyperinterpolation. Hyperinterpolation of degree $n$ is a discrete approximation of the $L^2$-orthogonal projection of degree $n$ with its Fourier coefficients evaluated by a positive-weight quadrature rule that exactly integrates all spherical polynomials of degree at most $2n$. This paper aims to bypass this quadrature exactness assumption by replacing it with the Marcinkiewicz--Zygmund property proposed in a previous paper. Consequently, hyperinterpolation can be constructed by a positive-weight quadrature rule (not necessarily with quadrature exactness). This scheme is referred to as unfettered hyperinterpolation. This paper provides a reasonable error estimate for unfettered hyperinterpolation. The error estimate generally consists of two terms: a term representing the error estimate of the original hyperinterpolation of full quadrature exactness and another introduced as compensation for the loss of exactness degrees. A guide to controlling the newly introduced term in practice is provided. In particular, if the quadrature points form a quasi-Monte Carlo (QMC) design, then there is a refined error estimate. Numerical experiments verify the error estimates and the practical guide.