Optimale Definite Polynome und Quadraturformeln

Summary. In this paper we study the remainder of interpolatory quadrature formulae. For this purpose we develop a simple but quite general comparison technique for linear functionals. Applied to quadrature formulae it allows to eliminate one of the nodes and to estimate the remainder of the old formula in terms of the new one. By repeated application we may compare with quadrature formulae having only a few nodes left or even no nodes at all. With the help of this method we obtain asymptotically best possible error bounds for the Clenshaw-Curtis quadrature and other Pdlya type formulae.Our comparison technique can also be applied to the problem of definiteness, i.e. the question whether the remainder R[f] of a formula of order m can be represented as c .fc,"!(~). By successive elimination of nodes we obtain a sequence of sufficient criteria for definiteness including all the criteria known to us as special cases. Finally we ask for good and worst quadrature formulae within certain classes. We shall see that amongst all quadrature formulae with positive coefficients and fixed order m the Gauss type formulae are worst. Interpreted in terms of Peano kernels our theorem yields results on monosplines which may be of interest in themselves.

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“…Locher [7] has shown that for a quadrature formula for which Steffensen's criterion applies we always have cm>~m-12 m(l+~l(l)) as m~c~.…”

Section: Is Definite Of Order M In Particular Positive (Or Negative)mentioning

confidence: 97%

“…Let n be even and let (1) be an interpolatory quadrature formula with symmetrically located nodes ordered as in (7). Let s be any integer satisfying n > s > n/2 + 1 and put…”

Section: Is Definite Of Order M In Particular Positive (Or Negative)mentioning

confidence: 99%

Error estimates for interpolatory quadrature formulae

Brass

Schmeißer

1981

Numer. Math.

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show abstract

“…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…”

Section: V~r)(x) = X P + ~(1 -X) Q + O W (X)mentioning

confidence: 99%

“…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.…”

Section: Obviouslymentioning

confidence: 99%