Numerical integration of H\"older continuous, absolutely convergent Fourier-, Fourier cosine-, and Walsh series

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 $\max ((s-1)N^{-1\mathrm{/}2},s^{\alpha \mathrm{/}2}{N}^{-\alpha })$. The construction of the quadrature points is explicit and is based on Weil sums.