Zur Struktur von Quadraturformeln

Abstract

Summary. Usually the error R n (f) of a quadrature formula is estimated with the aid or the Lz-norm of the Peano kernel. It is shown that this term may be estimated rather sharp using the norm I]Qnl[ of the quadrature rule. Then it follows that formulas with non-negative weights are favourable also in the sense of minimizing the Lx-norm of the kernel. A remainder term of the type R, (]) = c/(n+l) (~) is possible iff the kernel is definite. In the ease of an interpolatory formula this definiteness is usually shown by an application of the so-called "V-method". We determine the optimal formulas in the sense of this method. Then we analyse the influence of the structure of the mesh on the norm of a formula. We find that on an equidistant mesh with m nodes there exists a rule with a small norm if the order is not greater than cVm.

Optimal quadrature formula of Markov's and Locher's type with weight function

“…Let Pk(S't)(X) be a polynomial of degree k with leading coefficient 1, orthogonal respect to the weight function xS(1-x) z on [0, 1] to any polynomial of degree < k. Then from (25) and [1,2] we have…”

“…The formula (2) (according to [1,2]) is called the optimal formula in the meaning of V-method for the set c"+P+q+ 1 if the condition (10) In the following we will find such a formula.…”

“…where P,(x) is an arbitrary polynomial of degree <n. The last requirement is natural in view of condition (1). The restriction m<n<2m+l in case of given knots Xo,Xl,...,x m defines the coefficients of formula (2) (if the formula (2) exists) uniquely.…”

“…A quadrature formula of Markov's type with a weight function x~(1-x) ~, which has properties of formulas exact for polynomials of a given degree and properties of optimal formulas on some sets of functions, is given. The particular case of formula (where a=fl=p=q=O) is the formula of Locher [1,2]. …”