Numerical integration in log-Korobov and log-cosine spaces

Abstract

Quasi-Monte Carlo (QMC) rules are equal weight quadrature rules for approximating integrals over the s-dimensional unit cube [0, 1]s. One line of research studies the integration error of functions in the unit ball of so-called Korobov spaces, which are Hilbert spaces of periodic functions on [0, 1]s with square integrable partial mixed derivatives of order α. Using Parseval’s identity, this smoothness can be defined for all real numbers α > 1/2. In this setting, the condition α > 1/2 is necessary as otherwise the Korobov space contains discontinuous functions for which function evaluation is not well defined. This paper is concerned with more precise endpoint estimates of the integration error using QMC rules for Korobov spaces with α arbitrarily close to 1/2. To obtain such estimates we introduce a log-scale for function spaces with smoothness close to 1/2, which we call log-Korobov spaces. We show that lattice rules can be used to obtain an integration error of order \(\mathcal {O}(N^{-1/2} (\log N)^{-\mu (1-\lambda )/2})\) for any 1/μ < λ ≤ 1, where μ > 1 is a certain power in the log-scale. A second result is concerned with tractability of numerical integration for weighted Korobov spaces with product weights \((\gamma _{j})_{j \in \mathbb {N}}\). Previous results have shown that if \({\sum }_{j=1}^{\infty } \gamma _{j}^{\tau } < \infty \) for some 1/(2α) < τ ≤ 1 one can obtain error bounds which are independent of the dimension. In this paper we give a more refined estimate for the case where τ is close to 1/(2α), namely we show dimension independent error bounds under the condition that \({\sum }_{j=1}^{\infty } \gamma _{j} \max \{1, \log \gamma _{j}^{-1}\}^{\mu (1-\lambda )} < \infty \) for some 1/μ < λ ≤ 1. The essential tool in our analysis is a log-scale Jensen’s inequality.The results described above also apply to integration in log-cosine spaces using tent-transformed lattice rules.