Koksma–Hlawka type inequalities of fractional order

Abstract

The Koksma–Hlawka inequality states that the error of numerical integration by a quasi-Monte Carlo rule is bounded above by the variation of the function times the star-discrepancy. In practical applications though functions often do not have bounded variation. Hence here we relax the smoothness assumptions required in the Koksma–Hlawka inequality. We introduce Banach spaces of functions whose fractional derivative of order \({\alpha > 0}\) is in \({\mathcal{L}_p}\) . We show that if α is an integer and p = 2 then one obtains the usual Sobolev space. Using these fractional Banach spaces we generalize the Koksma–Hlawka inequality to functions whose partial fractional derivatives are in \({\mathcal{L}_p}\) . Hence we can also obtain an upper bound on the integration error even for certain functions which do not have bounded variation but satisfy weaker smoothness conditions.