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, whi...

We consider a positive measure on [0,∞) and a sequence of nested spaces L0 ⊂ L1 ⊂ L2 · · · of rational functions with prescribed poles in [−∞, 0]. Let {φk}k=0, with φ0 ∈ L0 and φk ∈ Lk \ Lk−1, k = 1, 2, . . . be the associated sequence of orthogonal rational functions. The zeros of φn can be used as the nodes of a rational Gauss quadrature formula that is exact for all functions in Ln · Ln−1, a...

In a previous paper it was shown how a probability measure (Eberlein measure) on the closed unit ball of the sequence space, h, can be used to find the variance a2 of the error functional for a quadrature formula for the fc-dimensional cube, regarded as a random variable. Here we give values of a for some specific formulae.

The approach we follow consists in transforming the numerical evaluation of hypersingular integrals into the calculation of a nearly singular integral whose mass is distributed according to a positive parameter ε. To evaluate the latter we apply a Gauss quadrature formula associated with a nearly singular weight function. It is estimated the error in terms of ε. Some numerical results are prese...