Information-based complexity
Tractability of Approximation for Weighted Korobov Spaces on Classical and Quantum Computers
Foundations of Computational Mathematics
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We study multivariate L"2-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 a={a"j} and b={b"j} of positive real numbers bounded away from zero. We study the minimal worst-case error e^L^"^2^-^a^p^p^,^@L(n,s) of all algorithms that use n information evaluations from the class @L in the s-variate case. We consider two classes @L in this paper: the class @L^a^l^l of all linear functionals and the class @L^s^t^d of only function evaluations. We study exponential convergence of the minimal worst-case error, which means that e^L^"^2^-^a^p^p^,^@L(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 weak, polynomial and strong polynomial tractability. In particular, 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 @L. They are also constructive with the exception of one particular sub-case for which we provide a semi-constructive algorithm.