Approximation of analytic functions in Korobov spaces

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 $e^{L_2-\mathrm{app},Λ}(n,s)$ be the minimal worst-case error of all algorithms that use $n$ information functionals from the class $Λ$ in the $s$-variate case. We consider two classes $Λ$: the class $Λ^{\rm all}$ consists of all linear functionals and the class $Λ^{\rm std}$ consists of only function valuations. We study (EXP) exponential convergence. This means that $$ e^{L_2-\mathrm{app},Λ}(n,s) \le C(s)\,q^{\,(n/C_1(s))^{p(s)}}\quad{for all}\quad n, s \in \mathbb{N} $$ 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 $e^{L_2-\mathrm{app},Λ}(n,s)\le \e$. 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 $c,τ_1,τ_2$ such that $n(\e,s)\le cs^{τ_1}(1+\log\,\e^{-1})^{τ_2}$ for all $s$ and $\e\in(0,1)$, and finally SPT holds iff the last estimate holds for $τ_1=0$. The infimum of $τ_2$ for which SPT holds is called the exponent $τ^*$ of SPT. We prove that the results are the same for both classes $Λ$, and obtain conditions for WT, PT, SPT with and without EXP and UEXP.