Numerical integration and discrepancy, a new approach

Abstract

The paper deals with two closely related topics: (i) Numerical integration of functions belonging to SobolevBesov spaces with dominating mixed smoothness in cubes. (ii) Discrepancy measured in terms of Lebesgue and Sobolev-Besov spaces.

Lp- and Sp,qrKritzinger1 2016Journal of Complexity6160View full textAdd to dashboardCite

“…There are also known lower and upper bounds for the S r p,q B-discrepancy in arbitrary dimensions. Triebel, who initiated the study of the local discrepancy in other spaces such as the Besov spaces and Triebel-Lizorkin spaces of dominating mixed smoothness in [31] and [32], showed that for all 1 ≤ p, q ≤ ∞ and r ∈ R satisfying 1 p − 1 < r < 1 p and q < ∞ if p = 1 and q > 1 if p = ∞ there exists a constant c 1 > 0 such that for any N ≥ 2 the local discrepancy of any N-element point set P in [0, 1) s satisfies…”

Discrepancy of Hammersley points in Besov spaces of dominating mixed smoothness

“…Apart from the L p -norms, until recently there was little done for other norms. In [17] and, in particular, in the forthcoming book [18] Triebel promoted the study of the discrepancy function in other function spaces such as suitable Sobolev, Besov or Triebel-Lizorkin spaces to gain more insight into its behavior and into applications to numerical integration. It is the purpose of this note to prove the conjectures in [18] concerning optimal upper bounds for the norm of the discrepancy function in Besov spaces of dominating mixed smoothness in the twodimensional case.…”

Proof techniques in quasi-Monte Carlo theory

“…Similarly, optimality can be shown in Besov spaces of dominating mixed smoothness for certain parameter values, as these can be characterized by an equivalent norm via Haar coefficients, see [37,94,95]. For generalizations to higher dimensions, see [58,59].…”