L∞-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 = { a j } and b = { b j } of positive real numbers bounded away from zero. We study the minimal worst-case error e L ∞ −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 Λ all of all linear functionals and the class Λ std of only function evaluations. We study exponential convergence of the minimal worst-case error, which means that e L ∞ −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 + log e − 1 to compute an e -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 . L 2 -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 L 2 -approximation. This allows us to deduce also results for L p -approximation for p ∈ [ 2 , ∞ ] .

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