We investigate quasi-Monte Carlo rules for the numerical integration of multivariate periodic functions from Besov spaces S r p,q B(T d ) with dominating mixed smoothness 1/ p < r < 2. We show that order 2 digital nets achieve the optimal rate of convergence N −r (log N ) (d−1)(1−1/q) . The logarithmic term does not depend on r and hence improves the known bound of Dick (SIAM J Numer Anal 45: 2007) for the special case of Sobolev spaces H r mix (T d ). Secondly, the rate of convergence is independent of the integrability p of the Besov space, which allows for sacrificing integrability in order to gain Besov regularity. Our method combines characterizations of periodic Besov spaces with dominating mixed smoothness via Faber bases with sharp estimates of Haar coefficients for the discrepancy function of order 2 digital nets. Moreover, we provide numerical computations which indicate that this bound also holds for the case r = 2.Mathematics Subject Classification 11K06 · 11J71 · 42C10 · 46E35 · 65C05 · 65D30 · 65D32 · 91G60

We prove sharp, two-sided bounds on sums of the form Sigma(d)(exp)(k epsilon N0)(Da(T))(-Sigma(d)(j=1) a(j)k(j)), where Da(T) := {k epsilon N-0(d) : Sigma(d)(j=1) a(j)k(j) <= T} and a epsilon R-+(d). These sums appear in the error analysis of tensor product approximation, interpolation and integration of d-variate analytic functions. Examples are tensor products of univariate Fourier-Legendre expansions (Beck et al., 2014) or interpolation and integration rules at Leja points (Chkifa et al., 2013), (Narayan and Jakeman, 2014), (Nobile et al., 2014). Moreover, we discuss the limit d -> infinity, where we prove both, algebraic and sub-exponential upper bounds. As an application we consider tensor products of Hardy spaces, where we study convergence rates of a certain truncated Taylor series, as well as of interpolation and integration using Leja points

We deal with lattices that are generated by the Vandermonde matrices associated to the roots of Chebyshev-polynomials. If the dimension d of the lattice is a power of two, i.e. d = 2 m , m ∈ N, the resulting lattice is an admissible lattice in the sense of Skriganov [12]. These are related to the Frolov cubature formulas, which recently drew attention due to their optimal convergence rates [18] in a broad range of Besov-Lizorkin-Triebel spaces. We prove that the resulting lattices are orthogonal and possess a lattice representation matrix with entries not larger than 2 (in modulus). This allows for an efficient enumeration of the Frolov cubature nodes in the d-cube [−1/2, 1/2] d up to dimension d = 16. *

We investigate quasi-Monte Carlo (QMC) integration of bivariate periodic functions with dominating mixed smoothness of order one. While there exist several QMC constructions which asymptotically yield the optimal rate of convergence ofit is yet unknown which point set is optimal in the sense that it is a global minimizer of the worst case integration error. We will present a computerassisted proof by exhaustion that the Fibonacci lattice is the unique minimizer of the QMC worst case error in periodic H 1 mix for small Fibonacci numbers N . Moreover, we investigate the situation for point sets whose cardinality N is not a Fibonacci number. It turns out that for N = 1, 2, 3, 5, 7, 8, 12, 13 the optimal point sets are integration lattices.

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