Multidimensional Quasi-Monte Carlo Integration in Weighted Anchored Sobolev Spaces

Abstract

In this article, an exact formula for the mean square worst-case error of the integration in the weighted anchored Sobolev spaces $$H_{\mathrm{Sob},s,\gamma,\mathbf{w}}$$ presented in terms of the functions of the system $$\Gamma_{\mathcal{B}_s}$$ is delivered. The notion of the so-called weighted anchored diaphony is introduced and is shown that it is a quantitative measure for the irregularity of the distribution of sequences. The relationship that exists between the mean square worst-case error and this type of the diaphony is established.