Cubature formulas, discrepancy, and nonlinear approximation

• Authors: V. N. Temlyakov
• Venue: Journal of Complexity
• Year: 2003

The main goal of this paper is to demonstrate connections between the following three big areas of research: the theory of cubature formulas (numerical integration), the discrepancy theory, and nonlinear approximation. In Section 1, we discuss a relation between results on cubature formulas and on discrepancy. In particular, we show how standard in the theory of cubature formulas settings can be translated into the discrepancy problem and into a natural generalization of the discrepancy problem. This leads to a concept of the r-discrepancy. In Section 2, we present results on a relation between construction of an optimal cubature formula with m knots for a given function class and best nonlinear m-term approximation of a special function determined by the function class. The nonlinear m-term approximation is taken with regard to a redundant dictionary also determined by the function class. Sections 3 and 4 contain some known results on the lower and the upper estimates of errors of optimal cubature formulas for the class of functions with bounded mixed derivative. One of the important messages (well known in approximation theory) of this paper is that the theory of discrepancy is closely connected with the theory of cubature formulas for the classes of functions with bounded mixed derivative. We have included in the paper both new results (Section 2) and known results. We included some known results with their proofs for the following two reasons. First of all we want to make the paper self-contained (within reasonable limits). Secondly, we selected the proofs which demonstrate different methods and are not very much technically involved. Section 5 contains historical notes on discrepancy and cubature formulas, some further comments and remarks. Historical remarks on nonlinear approximation are included in Section 2. We want to point out that this paper is not a complete survey in any of the above-mentioned areas. We did not even try to provide a complete list of results in those areas. We rather wanted to highlight the most typical results in cubature formulas (Sections 3 and 4) and show their relation to the discrepancy theory.