One Sided L1-Approximation and Bounds for Peano Kernels

Abstract

We investigate the Peano kernels of positive quadrature formulae. The kernels that we shall consider have a fixed argument, while the formulae have degree at least m, where m is some integer. We will show that there is a duality between these values and the error of polynomial, one sided, L1-approximation of maximal degree m. Using this duality, it will be possible to determine the set of quadrature formulae (extremal formulae) for which extreme values of Peano kernels are attained. Such results obviously yield error bounds for numerical integration; however, it is also possible to express the error constants for one sided L1-approximation in terms of quadrature errors. In some, special cases, the extremal formulae which generate an extremum of a Peano kernel at a certain point, are determined explicitly. In the final section, asymptotic bounds on Peano kernels, and hence asymptotic error constants in one sided approximation, are derived from the asymptotic estimates for the Peano kernels of the corresponding extremal formulae.