## Tractability of approximation for weighted Korobov spaces on classical and quantum computers

Venue: | Found. of Comput. Math |

Citations: | 11 - 1 self |

### BibTeX

@ARTICLE{Novak_tractabilityof,

author = {Erich Novak and Ian H. Sloan},

title = {Tractability of approximation for weighted Korobov spaces on classical and quantum computers},

journal = {Found. of Comput. Math},

year = {}

}

### OpenURL

### Abstract

We study the approximation problem (or problem of optimal recovery in the L2norm) for weighted Korobov spaces with smoothness parameter α. The weights γj of the Korobov spaces moderate the behavior of periodic functions with respect to successive variables. The non-negative smoothness parameter α measures the decay of Fourier coefficients. For α = 0, the Korobov space is the L2 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