Quadrature formulae of non-standard type

We discuss quadrature formulae of highest algebraic degree of precision for integration of functions of one or many variables which are based on non-standard data, i.e., in which the information used is different from the standard sampling of function values. Among the examples given in this survey is a quadrature formula for integration over the disk, based on linear integrals on n chords, which integrates exactly all bivariate algebraic polynomials of degree 2n− 1. 1 – Introduction The standard information used in univariate approximation methods consists of function values at a finite number of points. Important algorithms in numerical analysis dealing with recovery of functions, integrals, solutions of differential equations, zeros of functions and other quantities are based on data of function values. Most of the classical formulae are of this type. Famous examples are the Lagrange interpolation formula and the Newton-Cotes quadrature rules. It seems that, in the univariate case, sampling of function values is the most natural way of collecting information about a function. Besides, in practical problems, the standard outcome of experimental procedures and measurements is a function evaluation. That is why, studying approximation problems in the