$\boldsymbol{L}_{\infty}$-approximation in Korobov spaces with Exponential Weights

TL;DR

This paper investigates the exponential convergence and tractability of multivariate $L_$-approximation in weighted Korobov spaces with exponentially decaying Fourier coefficients, considering different information classes.

Contribution

It provides necessary and sufficient conditions on weights for exponential convergence and tractability in multivariate Korobov spaces, extending understanding of high-dimensional approximation.

Findings

Exponential convergence occurs under specific conditions on weight sequences.

Tractability notions depend on the decay rates of weights and are characterized precisely.

Results hold for both linear functional and standard information classes.

Abstract

We study multivariate $\boldsymbol{L}_{\infty}$-approximation for a weighted Korobov space of periodic functions for which the Fourier coefficients decay exponentially fast. The weights are defined, in particular, in terms of two sequences $\boldsymbol{a}=\{a_j\}$ and $\boldsymbol{b}=\{b_j\}$ of positive real numbers bounded away from zero. We study the minimal worst-case error $e^{\boldsymbol{L}_{\infty}\mathrm{-app},\Lambda}(n,s)$ of all algorithms that use $n$ information evaluations from a class $\Lambda$ in the $s$-variate case. We consider two classes $\Lambda$ in this paper: the class $\Lambda^{{\rm all}}$ of all linear functionals and the class $\Lambda^{{\rm std}}$ of only function evaluations. We study exponential convergence of the minimal worst-case error, which means that $e^{\boldsymbol{L}_{\infty}\mathrm{-app},\Lambda}(n,s)$ converges to zero exponentially fast with increasing $n$. Furthermore, we consider how the error depends on the dimension $s$. To this end, we define the notions of $\kappa$-EC-weak, EC-polynomial and EC-strong polynomial tractability, where EC stands for "exponential convergence". In particular, EC-polynomial tractability means that we need a polynomial number of information evaluations in $s$ and $1+\log\,\varepsilon^{-1}$ to compute an $\varepsilon$-approximation. We derive necessary and sufficient conditions on the sequences $\boldsymbol{a}$ and $\boldsymbol{b}$ for obtaining exponential error convergence, and also for obtaining the various notions of tractability. The results are the same for both classes $\Lambda$.