Approximation of analytic functions in Korobov spaces

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• 作者：Josef Dick^a ; ^{josef.dick@unsw.edu.au" class="auth_mail} ; Peter Kritzer^b ; ^{peter.kritzer@jku.at" class="auth_mail} ; Friedrich Pillichshammer^b ; ^{friedrich.pillichshammer@jku.at" class="auth_mail} ; Henryk Wo藕niakowski^c ; ^d ; ^{henryk@cs.columbia.edu" class="auth_mail}
• 关键词：Multivariate L2L2-approximation ; Worst-case error ; Tractability ; Exponential convergence ; Korobov space ; Quasi-Monte Carlo integration
• 刊名：Journal of Complexity
• 出版年：April, 2014
• 年：2014
• 卷：30
• 期：2
• 页码：2-28
• 全文大小：499 K

文摘

We study multivariate -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 and of positive real numbers bounded away from zero. We study the minimal worst-case error of all algorithms that use information evaluations from the class聽 in the -variate case. We consider two classes in this paper: the class of all linear functionals and the class of only function evaluations.

We study exponential convergence of the minimal worst-case error, which means that converges to zero exponentially fast with increasing . Furthermore, we consider how the error depends on the dimension . To this end, we define the notions of weak, polynomial and strong polynomial tractability. In particular, polynomial tractability means that we need a polynomial number of information evaluations in and to compute an -approximation. We derive necessary and sufficient conditions on the sequences and for obtaining exponential error convergence, and also for obtaining the various notions of tractability. The results are the same for both classes . They are also constructive with the exception of one particular sub-case for which we provide a semi-constructive algorithm.