On a generalization of compound Newton-Cotes quadrature formulas

Abstract

We consider quadrature formulas defined by piecewise polynomial interpolation at equidistant nodes, admitting the nodes of adjacent polynomials to overlap, which generalizes the interpolation scheme of the compound Newton-Cotes quadrature formulas. The error constantse μ,n in the estimate $$|R_n [f]| \leqslant e_{\mu ,n} ||f^{(\mu )} ||_\infty$$ are considered for the highest possible values of μ, which are μ=r ifr is even, and μ=r+1 ifr is odd (wherer − 1 is the degree of the polynomials used for interpolation). It is determined which quadrature formulas of the type introduced have (asymptotically) the least error constant. As a result, though the compound Newton-Cotes quadrature formulas have an optimality property, they are not the best formulas of this type.