Quadraturrest, Approximation und Chebyshev-Polynome

Abstract

The remainder of a quadrature formula is often treated by introducing derivatives of high order. But in many cases it is preferable to avoid derivatives and to use more robust methods. One can consider series expansions (Hilbert space, holomorphy). But there are simpler methods, employing polynomials, approximation, grids. In connection with quadrature such methods have been worked out by several authors; we mention Stroud, Locher-Zeller, Riess-Johnson and especially the recent book by Brass. In the present note we continue these investigations. We utilize approximation degrees of different order (in connection with Chebyshev expansions) and L-approximation. The general results are applied to the quadrature schemes of Clenshaw-Curtis and of Filippi. An outlook mentions extensions, cubature and fast approximation procedures.