Hyperinterpolation beyond exact cubature: a spectral multiplier approach
For researchers in approximation theory and numerical analysis, this provides a unified framework for stable scattered data approximation without exact polynomial reproduction, extending the applicability of hyperinterpolation.
The paper develops a theory of hyperinterpolation on the sphere that relaxes the need for exact polynomial cubature formulas, using spectral multiplier operators to handle discretization error. This yields stable Sobolev approximation estimates under weak cubature assumptions and connects hyperinterpolation to Sobolev discrepancy and quasi-Monte Carlo designs.
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.