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We introduce quasi-Monte Carlo rules for the numerical integration offunctions $f$ defined on $[0,1]^s$, $s \ge 1$, which satisfy the followingproperties: the Fourier-, Fourier cosine- or Walsh coefficients of $f$ areabsolutely 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 errorof order $\max((s-1) N^{-1/2}, s^{\alpha/2} N^{-\alpha} )$. The construction ofthe quadrature points is explicit and is based on Weil sums.

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