Tractability of Approximation for Weighted Korobov Spaces on Classical and Quantum Computers

摘要

We study the approximation problem (or problem of optimal recovery in the L_2-norm) for weighted Korobov spaces with smoothness parameter α. The weights γ_j of the Korobov spaces moderate the behavior of periodic functions with respect to successive variables. The nonnegative smoothness parameter α measures the decay of Fourier coefficients. For α=0, the Korobov space is the L_2 space, whereas for positive α, the Korobov space is a space of periodic functions with some smoothness and the approximation problem corresponds to a compact operator. The periodic functions are defined on [0,1]~d and our main interest is when the dimension d varies and may be large. We consider algorithms using two different classes of information. The first class Λ~(all) consists of arbitrary linear functionals. The second class Λ~(std) consists of only function values and this class is more realistic in practical computations. We want to know when the approximation problem is tractable. Tractability means that there exists an algorithm whose error is at most ε and whose information cost is bounded by a polynomial in the dimension d and in ε~(-1). Strong tractability means that the bound does not depend on d and is polynomial in ε~(-1). In this paper we consider the worst case, randomized, and quantum settings. In each setting, the concepts of error and cost are defined differently and, therefore, tractability and strong tractability depend on the setting and on the class of information. In the worst case setting, we apply known results to prove that strong tractability and tractability in the classre equivalent.