Hyperinterpolation beyond exact cubature: a spectral multiplier approach

TL;DR

This paper introduces a spectral multiplier approach to hyperinterpolation on the sphere that relaxes the need for exact cubature formulas, enabling stable approximation from scattered data under weak assumptions.

Contribution

It develops a new framework interpreting discretization error as a spectral multiplier operator, broadening hyperinterpolation theory beyond classical exact cubature requirements.

Findings

Stable Sobolev approximation estimates under weak cubature assumptions.

Applicability to various spectral approximation operators including heat kernels.

Uniform $L^ Infty$-stability for localized spectral multipliers.

Abstract

We study hyperinterpolation and its spectral multiplier variants on the sphere under weak cubature assumptions formulated through Sobolev discrepancy estimates. In contrast with classical hyperinterpolation theory, our framework does not require exact polynomial cubature formulas or Marcinkiewicz--Zygmund inequalities. The main idea is to interpret the discretization error as the action of a spectral multiplier operator on the cubature discrepancy measure. This viewpoint separates approximation properties of the underlying spectral operator from geometric properties of the sampling measure, leading to stable Sobolev approximation estimates under weak cubature assumptions. The resulting theory applies to a broad class of spectral approximation operators, including sharp spectral projections, compactly supported smooth filters, Bessel potential operators, and heat kernel operators. For sufficiently localized spectral multipliers, we additionally obtain uniform $L^\infty$-stability of the corresponding discrete approximation operators. The results establish a direct connection between hyperinterpolation, Sobolev discrepancy, and quasi-Monte Carlo (QMC) designs, showing that stable approximation from scattered data can be achieved without exact polynomial reproduction.