Spezielle interpolationsquadraturen vom gauss'schen typ

Abstract

GAUSS' quadrature procedure, based upon the zeros of LEGENDRE-polynomials as nodes, may be also developed by integration of an HERMITE-interpolation-polynomial involving first derivatives of the integrand at the same nodes. All the weights of these derivatives in the resulting quadrature formula vanish, while the weights of the function values are positive. After building up a more general HERMITE-interpolation-operator (cf. |6|, |8|, |10|) general quadratures are obtained by integration. They are defined as general GAUSS-type-quadratures if and only if all weights of the derivatives of the quadrature vanish. As a consequence the weights of the function values of such a quadrature are shown to be positive. Besides the ordinary GAUSS-quadrature and variants with weight-functions within the integrand there ist not only the possibility of constructing arbitrarily sets of new GAUSS-type-quadratures but also some known quadrature-procedures are shown to be of this type, e. g. WILF-quadrature (cf. |18|, |4|) and a procedure due to MARTENSEN (cf. |15|). For the WILF-quadrature new aspects for the numerical determination of nodes and weights follow from this new point of view. It is well known that this problem is very ill conditioned. The usual GAUSS-quadrature surely is contained as a special case for the various weight-functions. It should be mentioned, that the general interpolating operator yields also interesting results in numerical differentiation (cf. |9|).