ERROR-ESTIMATES FOR GENERALIZED COMPOUND QUADRATURE-FORMULAS

We define quadrature formulas for integrals with weight functions by applying a given approximation method locally. This allows the generalisation of different quadrature formulas, e.g., the compound Newton-Cotes formulas, Gauss summation formulas, or Gregory's formulas, to the case of weighted integrals, as well as to construct new quadrature formulas, and to derive error estimates for all these quadrature formulas. The estimates considered here are mainly of the form \R[f]\ less-than-or-equal-to c \\f(r)\\, provided the underlying approximation method is exact for polynomials of degree < r (R[f] is the quadrature error). Explicit, asymptotically sharp error estimates are obtained for arbitrary integrable weight functions. Further, estimates are obtained for the case that the quadrature error is of higher order than the approximation error.