On the quadrature exactness in hyperinterpolation

TL;DR

This paper explores how relaxing the quadrature exactness requirement in hyperinterpolation affects approximation quality, showing that convergence can still be achieved with fewer points, supported by theoretical analysis and numerical experiments.

Contribution

It demonstrates that hyperinterpolation remains convergent under relaxed quadrature exactness, broadening the choice of quadrature rules and reducing computational complexity.

Findings

Convergence is maintained when quadrature exactness is relaxed proportionally to polynomial degree.

The $L^2$ norm of the hyperinterpolation operator remains bounded independently of $n$.

Numerical experiments confirm theoretical predictions using Gauss, Clenshaw--Curtis, and spherical t-design quadratures.

Abstract

This paper investigates the role of quadrature exactness in the approximation scheme of hyperinterpolation. Constructing a hyperinterpolant of degree $n$ requires a positive-weight quadrature rule with exactness degree $2n$. We examine the behavior of such approximation when the required exactness degree $2n$ is relaxed to $n+k$ with $0<k\leq n$. Aided by the Marcinkiewicz--Zygmund inequality, we affirm that the $L^2$ norm of the exactness-relaxing hyperinterpolation operator is bounded by a constant independent of $n$, and this approximation scheme is convergent as $n\rightarrow\infty$ if $k$ is positively correlated to $n$. Thus, the family of candidate quadrature rules for constructing hyperinterpolants can be significantly enriched, and the number of quadrature points can be considerably reduced. As a potential cost, this relaxation may slow the convergence rate of hyperinterpolation in terms of the reduced degrees of quadrature exactness. Our theoretical results are asserted by numerical experiments on three of the best-known quadrature rules: the Gauss quadrature, the Clenshaw--Curtis quadrature, and the spherical $t$-designs.