Bypassing the quadrature exactness assumption of hyperinterpolation on the sphere

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.

Certified reduced order method for the parametrized Allen-Cahn equation

“…In [40], the construction of hyperinterpolation relies on the quadrature exactness (1.9). However, recent works in [3,4] has relaxed and even bypassed this assumption. In this paper, the quadrature exactness (1.9) is not a necessary assumption for our scheme; we only make the following three natural and simple assumptions: Assumption 1.1 For the quadrature rule (1.8), we assume that (I) it integrates all constants exactly; namely, m j=1 w…”