The main goal of this talk 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. First, I will discuss a relation between results on cubature formulas and on discrepancy. In particular, I'll 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. Second, I'll 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. Third, I'll give 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 talk is that the theory of discrepancy is closely connected with the theory of cubature formulas for the classes of functions with bounded mixed derivative.
