2016/02/08 by Peter Kritzer, Kritzer, Peter, Friedrich Pillichshammer +3
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical functions and polynomials #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1602.02572
openalex publication_date 2016/02/08 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/01
We study multivariate \boldsymbolL∞-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 \boldsymbola=\aj\ and \boldsymbolb=\bj\ of positive real numbers bounded away from zero. We study the minimal worst-case error e^\boldsymbolL∞-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\rm all of all linear functionals and the class Λ\rm std of only function evaluations. We study exponential convergence of the minimal worst-case error, which means that e^\boldsymbolL∞-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 \boldsymbola and \boldsymbolb for obtaining exponential error convergence, and also for obtaining the various notions of tractability. The results are the same for both classes Λ.