Numerical weighted integration of functions having mixed smoothness

Change of variable in spaces of mixed smoothness and numerical integration of multivariate functions on the unit cube

In a recent article by two of the present authors it turned out that Frolov’s cubature formulae are optimal and universal for various settings (Besov-Triebel-Lizorkin spaces) of functions with dominating mixed smoothness. Those cubature formulae go well together with functions supported inside the unit cube [0, 1]. The question for the optimal numerical integration of multivariate functions wit...

A Numerical Integration Method by Using Generalized Series of Functions

Quasi-Monte Carlo methods for integration of functions with dominating mixed smoothness in arbitrary dimension

In a celebrated construction, Chen and Skriganov gave explicit examples of point sets achieving the best possible L2-norm of the discrepancy function. We consider the discrepancy function of the ChenSkriganov point sets in Besov spaces with dominating mixed smoothness and show that they also achieve the best possible rate in this setting. The proof uses a b-adic generalization of the Haar syste...

Weighted moduli of smoothness of k-monotone functions and applications

Let ωk φ( f, δ)w,Lq be the Ditzian–Totik modulus with weight w, M k be the cone of k-monotone functions on (−1, 1), i.e., those functions whose kth divided differences are nonnegative for all selections of k + 1 distinct points in (−1, 1), and denote E(X, Pn)w,q := sup f ∈X infP∈Pn ∥w( f − P)∥Lq , where Pn is the set of algebraic polynomials of degree at most n. Additionally, let wα,β (x) := (1...

Asymptotically Optimal Weighted Numerical Integration

We study numerical integration of H older{type functions with respect to weights on the real line. Our study extends previous work by F. Curbera, [2] and relies on a connection between this problem and the approximation of distribution functions by empirical ones. The analysis is based on a lemma which is important within the theory of optimal designs for approximating stochastic processes. As...