Optimal point sets for quasi-Monte Carlo integration of bivariate periodic functions with bounded mixed derivatives

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 of $\mathcal{O}(N^{-1}\log(N)^{\frac{1}{2}})$, it 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 computer-assisted proof by exhaustion that the Fibonacci lattice is the unique minimizer of the QMC worst case error in periodic $H^1_\text{mix}$ for small $N$. Moreover, we investigate the situation for pointsets 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.Comment: 20 pages, version 2: several minor changes incorporating referee's suggestion

Johnson-Lindenstrauss lemma for circulant matrices

We prove a variant of a Johnson-Lindenstrauss lemma for matrices with circulant structure. This approach allows to minimise the randomness used, is easy to implement and provides good running times. The price to be paid is the higher dimension of the target space $k=O(\epsilon^{-2}\log^3n)$ instead of the classical bound $k=O(\epsilon^{-2}\log n)$

The average distance property of classical Banach spaces II

A Banach space X has the average distance property (ADP) if there exists a unique real number r such that for each positive integer n and all x_1,...,x_n in the unit sphere of X there is some x in the unit sphere of X such that 1/n \sum_{k=1}^n ||x_k-x|| = r. We show that l_p does not have the average distance property if p>2. This completes the study of the ADP for l_p spaces.Comment: 10 pages, to appear in Bull. Austr. Math. So

Fibonacci lattices have minimal dispersion on the two-dimensional torus

We study the size of the largest rectangle containing no point of a given point set in the two-dimensional torus, the dispersion of the point set. A known lower bound for the dispersion of any point set of cardinality $n\ge 2$ in this setting is $2/n$. We show that if $n$ is a Fibonacci number then the Fibonacci lattice has dispersion exactly $2/n$ meeting the lower bound. Moreover, we completely characterize integration lattices achieving the lower bound and provide insight into the structure of other optimal sets. We also treat related results in the nonperiodic setting.Comment: 13 page

Cubature Formulas for Symmetric Measures in Higher Dimensions with Few Points

We study cubature formulas for d-dimensional integrals with an arbitrary symmetric weight function of tensor product form. We present a construction that yields a high polynomial exactness: for fixed degree l=5 or l=7 and large dimension, the number of knots is only slightly larger than the lower bound of M\"oller and much smaller compared to the known constructions. We also show, for any odd degree l=2k+1, that the minimal number of points is almost independent of the weight function. This is also true for the integration over the (Euclidean) sphere.Comment: 19 pages, Mathematics of Computation, to appea

An improved lower bound for the L2L_2-discrepancy

We give an improved lower bound for the $L_2$-discrepancy of finite point sets in the unit square.Comment: 11 pages, version 2: several minor changes incorporating referee's suggestion

On the randomized complexity of Banach space valued integration

We study the complexity of Banach space valued integration in the randomized setting. We are concerned with $r$-times continuously differentiable functions on the $d$-dimensional unit cube $Q$, with values in a Banach space $X$, and investigate the relation of the optimal convergence rate to the geometry of $X$. It turns out that the $n$-th minimal errors are bounded by $cn^{-r/d-1+1/p}$ if and only if $X$ is of equal norm type $p$.Comment: 12 page

On the Non-Equivalence of Rearranged Walsh and Trigonometric Systems in L_p

We consider the question whether the trigonometric system can be equivalent to some rearrangement of the Walsh system in L_p for some p2. We show that this question is closely related to a combinatorial problem. This enables us to prove non-equivalence for a number of rearrangements. Previously this was known for the Walsh-Paley order only.Comment: 18 pages, to be published in Stud. Mat

Entropy numbers of spheres in Banach and quasi-Banach spaces

We prove sharp upper bounds on the entropy numbers $e_k(S^{d-1}_p,\ell_q^d)$ of the $p$-sphere in $\ell_q^d$ in the case $k \geq d$ and $0< p \leq q \leq \infty$. In particular, we close a gap left open in recent work of the second author, T. Ullrich and J. Vybiral. We also investigate generalizations to spheres of general finite-dimensional quasi-Banach spaces

Optimal order of LpL_p-discrepancy of digit shifted Hammersley point sets in dimension 2

It is well known that the two-dimensional Hammersley point set consisting of $N=2^n$ elements (also known as Roth net) does not have optimal order of $L_p$-discrepancy for $p \in (1,\infty)$ in the sense of the lower bounds according to Roth (for $p \in [2,\infty)$) and Schmidt (for $p \in (1,2)$). On the other hand, it is also known that slight modifications of the Hammersley point set can lead to the optimal order $\sqrt{\log N}/N$ of $L_2$-discrepancy, where $N$ is the number of points. Among these are for example digit shifts or the symmetrization. In this paper we show that these modified Hammersley point sets also achieve optimal order of $L_p$-discrepancy for all $p \in (1,\infty)$