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

Abstract

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 Hölder condition of order $α$, for some $0 < α\le 1$. We show a convergent rate of the integration error of order $\max((s-1) N^{-1/2}, s^{α/2} N^{-α} )$. The construction of the quadrature points is explicit and is based on Weil sums.