Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces

Abstract

We introduce a weighted reproducing kernel Hilbert space which is based on Walsh functions. The worst-case error for integration in this space is studied, especially with regard to $(t,m,s)$-nets. It is found that there exists a digital $(t,m,s)$-net, which achieves a strong tractability worst-case error bound under certain condition on the weights.

We also investigate the worst-case error of integration in weighted Sobolev spaces. As the main tool we define a digital shift invariant kernel associated to the kernel of the weighted Sobolev space. This allows us to study the mean square worst-case error of randomly digitally shifted digital $(t,m,s)$-nets. As this digital shift invariant kernel is almost the same as the kernel for the Hilbert space based on Walsh functions, we can derive results for the weighted Sobolev space based on the analysis of the Walsh function space. We show that there exists a $(t,m,s)$-net which achieves the best possible convergence order for integration in weighted Sobolev spaces and are strongly tractable under the same condition on the weights as for lattice rules.