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 worstcase 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 ε −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.L 2 -approximation for functions from the same function space has been considered in [2]. 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, ∞].

RICAM, the Johann Radon Institute for Computational and Applied Mathematics

“…It is also shown there that for L 2 -approximation all results are the same independently of whether we consider information from Λ all or Λ std , which is rather surprising. Moreover, it was shown in [23] that almost all conditions remain the same if we consider L ∞ -approximation instead of L 2 -approximation for Λ all or Λ std , and also in the case of L ∞approximation all results for Λ all and Λ std coincide. For results on numerical integration in H smooth kor,d,a,b , we refer to [22].…”

“…Indeed, the study of multivariate problems for spaces of infinitely smooth functions motivated the introduction of exponential tractability. Motivating examples with positive exponential tractability results may be found for specific spaces in the papers [1,5,16,22,23,26,27,43] and the references therein.…”

Exponential tractability of linear weighted tensor product problems in the worst-case setting for arbitrary linear functionals

“…Taking the logarithm we obtain This implies a c j ≤ 1 j 2 for all j ≥ j c . Hence (11) holds. ✷ Lemma 3.…”

“…For results on exponential tractability for general linear problems (without necessarily assuming tensor product structure), we refer the reader to the recent paper [12], for exponential tractability for linear problems on un-weighted tensor product spaces, we refer the reader to [4,21,22]. Further results on exponential tractability can, e.g., be found in the papers [1,2,6,7,8,10,11,13,14,15,16,23,27,28,31,33].…”

Tractability of approximation in the weighted Korobov space in the worst-case setting

“…Furthermore, it is well known, see, e.g., [3], that L 2 -approximation is not harder than L ∞ -approximation for the absolute error criterion, which means that for Λ ∈ {Λ all , Λ std } we have…”