Approximation of analytic functions in Korobov spaces

TL;DR

This paper investigates the exponential convergence rates of multivariate approximation in Korobov spaces of analytic periodic functions, analyzing how different classes of algorithms and tractability notions affect approximation efficiency.

Contribution

It provides new conditions under which exponential and uniform exponential convergence, along with various tractability properties, hold for multivariate approximation in Korobov spaces.

Findings

Exponential convergence (EXP) and uniform exponential convergence (UEXP) are characterized.

Conditions for weak, polynomial, and strong polynomial tractability are established.

Results are consistent across all linear and standard information classes.

Abstract

We study multivariate $L_2$-approximation for a weighted Korobov space of analytic 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 numbers no less than one. Let $e^{L_2-\mathrm{app},\Lambda}(n,s)$ be the minimal worst-case error of all algorithms that use $n$ information functionals from the class $\Lambda$ in the $s$-variate case. We consider two classes $\Lambda$: the class $\Lambda^{{\rm all}}$ consists of all linear functionals and the class $\Lambda^{{\rm std}}$ consists of only function valuations. We study (EXP) exponential convergence. This means that $$ e^{L_2-\mathrm{app},\Lambda}(n,s) \le C(s)\,q^{\,(n/C_1(s))^{p(s)}}\quad{for all}\quad n, s \in \mathbb{N} $$ where $q\in(0,1)$, and $C,C_1,p:\mathbb{N} \rightarrow (0,\infty)$. If we can take $p(s)=p>0$ for all $s$ then we speak of (UEXP) uniform exponential convergence. We also study EXP and UEXP with (WT) weak, (PT) polynomial and (SPT) strong polynomial tractability. These concepts are defined as follows. Let $n(\e,s)$ be the minimal $n$ for which $e^{L_2-\mathrm{app},\Lambda}(n,s)\le \e$. Then WT holds iff $\lim_{s+\log\,\e^{-1}\to\infty}(\log n(\e,s))/(s+\log\,\e^{-1})=0$, PT holds iff there are $c,\tau_1,\tau_2$ such that $n(\e,s)\le cs^{\tau_1}(1+\log\,\e^{-1})^{\tau_2}$ for all $s$ and $\e\in(0,1)$, and finally SPT holds iff the last estimate holds for $\tau_1=0$. The infimum of $\tau_2$ for which SPT holds is called the exponent $\tau^*$ of SPT. We prove that the results are the same for both classes $\Lambda$, and obtain conditions for WT, PT, SPT with and without EXP and UEXP.