Discrepancy and integration in function spaces with dominating mixed smoothness

TL;DR

This paper analyzes discrepancy bounds in Besov, Triebel-Lizorkin, and Sobolev spaces with dominating mixed smoothness, extending known results to higher dimensions and broader classes of point sets, and examining implications for numerical integration.

Contribution

It provides new upper bounds for discrepancy in higher dimensions, characterizes $b$-adic point sets in these spaces, and extends results to Triebel-Lizorkin and Sobolev spaces.

Findings

Larger class of point sets satisfying optimal upper bounds.

Extension of discrepancy bounds to arbitrary dimensions.

Results for integration error in related function spaces.

Abstract

Optimal lower bounds for discrepancy in Besov spaces with dominating mixed smoothness are known from the work of Triebel. Hinrichs proved upper bounds in the plane. In this work we systematically analyse the problem, starting with a survey of discrepancy results and the calculation of the best known constant in Roth's Theorem. We give a larger class of point sets satisfying the optimal upper bounds than already known from Hinrichs for the plane and solve the problem in arbitrary dimension for certain parameters considering a celebrated constructions by Chen and Skriganov which are known to achieve optimal $L_2$-norm of the discrepancy function. Since those constructions are $b$-adic, we give $b$-adic characterizations of the spaces. Finally results for Triebel-Lizorkin and Sobolev spaces with dominating mixed smoothness and for the integration error are concluded.