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We investigate quasi-Monte Carlo rules for the numerical integration ofmultivariate periodic functions from Besov spaces $S^r_{p,q}B(\mathbb{T}^d)$with dominating mixed smoothness $1/p<r<2$. We show that order 2 digital netsachieve the optimal rate of convergence $N^{-r} (\log N)^{(d-1)(1-1/q)}$. Thelogarithmic term does not depend on $r$ and hence improves the known boundprovided by J. Dick for the special case of Sobolev spaces$H^r_{\text{mix}}(\mathbb{T}^d)$. Secondly, the rate of convergence isindependent of the integrability $p$ of the Besov space, which allows forsacrificing integrability in order to gain Besov regularity. Our methodcombines characterizations of periodic Besov spaces with dominating mixedsmoothness via Faber bases with sharp estimates of Haar coefficients for thediscrepancy function of higher order digital nets. Moreover, we providenumerical computations which indicate that this bound also holds for the case$r=2$.

Optimal quasi-Monte Carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions