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Koksma-Hlawka type inequalities of fractional order. (English) Zbl 1223.11095

Summary: 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 \(\alpha\) 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.

MSC:

11K38 Irregularities of distribution, discrepancy
65D30 Numerical integration
26A33 Fractional derivatives and integrals
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