Error bounds for function approximation using generated sets

TL;DR

This paper establishes error bounds for function approximation in reproducing kernel Hilbert spaces using generated sets, demonstrating optimal convergence and tractability in specific spaces like weighted Korobov spaces.

Contribution

It introduces a novel analysis of generated sets for function approximation, showing optimal error bounds and tractability results in certain function spaces.

Findings

Existence of generators with optimal convergence rates.

Rational generators can be used with slight bound increases.

Derived bounds for weighted Korobov spaces with few samples.

Abstract

This paper explores the use of "generated sets" $\{ \{ k \boldsymbol{\zeta} \} : k = 1, \ldots, n \}$ for function approximation in reproducing kernel Hilbert spaces which consist of multi-dimensional functions with an absolutely convergent Fourier series. The algorithm is a least squares algorithm that samples the function at the points of a generated set. We show that there exist $\boldsymbol{\zeta} \in [0,1]^d$ for which the worst-case $L_2$ error has the optimal order of convergence if the space has polynomially converging approximation numbers. In fact, this holds for a significant portion of the generators. Additionally we show that a restriction to rational generators is possible with a slight increase of the bound. Furthermore, we specialise the results to the weighted Korobov space, where we derive a bound applicable to low values of sample points, and state tractability results.