Lattice algorithms for multivariate approximation in periodic spaces with general weight parameters

TL;DR

This paper develops lattice algorithms for multivariate periodic function approximation in weighted spaces, achieving optimal convergence rates and extending applicability to PDE-related problems with complex weight parameters.

Contribution

It introduces a theoretical framework for lattice algorithms in weighted periodic spaces with general weights, surpassing previous methods limited to product weights.

Findings

Achieves optimal convergence rates for lattice algorithms in weighted spaces.

Extends applicability to PDE problems with POD and SPOD weights.

Provides a foundation for lattice-based approximation methods in complex weighted settings.

Abstract

This paper provides the theoretical foundation for the construction of lattice algorithms for multivariate $L_2$ approximation in the worst case setting, for functions in a periodic space with general weight parameters. Our construction leads to an error bound that achieves the best possible rate of convergence for lattice algorithms. This work is motivated by PDE applications in which bounds on the norm of the functions to be approximated require special forms of weight parameters (so-called POD weights or SPOD weights), as opposed to the simple product weights covered by the existing literature. Our result can be applied to other lattice-based approximation algorithms, including kernel methods or splines.