Das asymptotische Verhalten der Peanokerne einiger interpolatorischer Quadraturverfahren

Summary

Interpolatory quadrature formulae consist in replacing$\int\limits_{ - 1}^1 {f(x) dx}$ by$\int\limits_{ - 1}^1 {p_f (x) dx}$ wherep _f denotes the interpolating polynomial off with respect to a certain knot setX. The remainder$R(f) = \int\limits_{ - 1}^1 {(f(x) - p_f (x)) dx}$ may in many cases be written as$\int\limits_{ - 1}^1 {P_X (t)f^{(m)} (t) dt}$ wherem=n resp. (n+1) forn even and odd, respectively. We determine the asymptotic behaviour of the Peano kernelP _X (t) forn→∞ for the quadrature formulae of Filippi, Polya and Clenshaw-Curtis.