Formula omitted]-Approximation in Korobov spaces with exponential weights

Abstract

We study multivariate L∞-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 a={aj} and b={bj} of positive real numbers bounded away from zero. We study the minimal worst-case error eL∞−app,Λ(n,s) of all algorithms that use n information evaluations from a class Λ in the s-variate case. We consider two classes Λ in this paper: the class Λall of all linear functionals and the class Λstd of only function evaluations.We study exponential convergence of the minimal worst-case error, which means that eL∞−app,Λ(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 κ-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+logε−1 to compute an ε-approximation. We derive necessary and sufficient conditions on the sequences a and b for obtaining exponential error convergence, and also for obtaining the various notions of tractability. The results are the same for both classes Λ.L2-approximation for functions from the same function space has been considered in Dick et al. (2014). It is surprising that most results for L∞-approximation coincide with their counterparts for L2-approximation. This allows us to deduce also results for Lp-approximation for p∈[2,∞].