The theory of quadratures -- concepts and results (Q1581023)

By quoting selected theorems and definitions, the author gives an overview on recent developments in the numerical computation of integrals. The focus of the article are both theoretical foundations and concrete algorithms. Introducing a basic principle, the author defines the co-observation of a numerical integration problem as the information on the integrand function \(f\) which limits \(f\) at points where no function value has been determined. A co-observation defines a class of functions and thus leads to the concept of optimality. The author quotes deep and important theorems on optimal quadrature formulas for several function spaces. Observing that a function, in general, belongs to a multitude of function spaces, the author discusses the question of choosing a ``good'' class and corresponding optimal formula. A simpler guiding principle, given in section 3, is to construct algorithms such that are exact for a given space of functions. The Gaussian quadrature formula and its important generalization for weak Chebeshev systems [cf. \textit{C. A. Micchelli} and \textit{A. Pinkus}, SIAM J. Math. Anal. 8, 206-230 (1977; Zbl 0353.41003)] are prominent examples. For quadrature formulas constructed from this guiding principle, error bounds can easily be deduced from Lebesgue's inequality and results from approximation theory. Several results on norms of quadrature formulas are discussed. The article concludes with a discussion of results on error bounds and error asymptotics for quadrature formulas, stating also an open problem for the trapezoidal rule.