Numerical integration of Hölder continuous, absolutely convergent Fourier-, Fourier cosine-, and Walsh series

Josef Dick

Published 2013-12-04, updated 2014-03-12Version 2

We introduce quasi-Monte Carlo rules for the numerical integration of functions $f$ defined on $[0,1]^s$, $s \ge 1$, which satisfy the following properties: the Fourier-, Fourier cosine- or Walsh coefficients of $f$ are absolutely summable and $f$ satisfies a H\"older condition of order $\alpha$, for some $0 < \alpha \le 1$. We show a convergent rate of the integration error of order $\max((s-1) N^{-1/2}, s^{\alpha/2} N^{-\alpha} )$. The construction of the quadrature points is explicit and is based on Weil sums.