An Explicit Representation of the Remainder of some Newton-Cotes Formulas in Terms of Higher Order Differences

Abstract

Depending upon the exactness of the rule, the remainders of some Newton-Cotes formulas are explicitly represented in terms of higher order differences. Consequently, those error bounds for the associated compound quadrature processes, given via corresponding moduli of continuity, may now beestablished in a completely elementary way, in fact with good constants. As an application of previous quantitative extensions of the uniform boundedness principle it is finally shown that the error estimates considered are always sharp.