منابع مشابه
On Stieltjes Polynomials and Gauss-kronrod Quadrature
Let D be a real function such that D(z) is analytic and D(z) ± 0 for \z\ < 1. Furthermore, put W(x) = \J\ x2\D(e'v)\2 , x = costp , tp e [0, 71 ], and denote by pn(', rV) the polynomial which is orthogonal on [-1, +1] to Pn_[ (P„_! denotes the set of polynomials of degree at most n 1 ) with respect to W . In this paper it is shown that for each sufficiently large n the polynomial En+X(-, W) (ca...
متن کاملStieltjes Polynomials and the Error of Gauss-kronrod Quadrature Formulas
The Gauss-Kronrod quadrature scheme, which is based on the zeros of Legendre polynomials and Stieltjes polynomials, is the standard method for automatic numerical integration in mathematical software libraries. For a long time, very little was known about the error of the Gauss-Kronrod scheme. An essential progress was made only recently, based on new bounds and as-ymptotic properties for the S...
متن کاملStieltjes polynomials and related quadrature formulae for a class of weight functions
Consider a (nonnegative) measure dσ with support in the interval [a, b] such that the respective orthogonal polynomials, above a specific index `, satisfy a three-term recurrence relation with constant coefficients. We show that the corresponding Stieltjes polynomials, above the index 2` − 1, have a very simple and useful representation in terms of the orthogonal polynomials. As a result of thi...
متن کاملStieltjes polynomials and Lagrange interpolation
Bounds are proved for the Stieltjes polynomial En+1, and lower bounds are proved for the distances of consecutive zeros of the Stieltjes polynomials and the Legendre polynomials Pn. This sharpens a known interlacing result of Szegö. As a byproduct, bounds are obtained for the Geronimus polynomials Gn. Applying these results, convergence theorems are proved for the Lagrange interpolation process...
متن کاملFull Quadrature Sums for Generalized Polynomials with Freud Weights
Generalized nonnegative polynomials are defined as products of nonnegative polynomials raised to positive real powers. The generalized degree can be defined in a natural way. In this paper we extend quadrature sums involving pth powers of polynomials to those for generalized polynomials.
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ژورنال
عنوان ژورنال: Constructive Approximation
سال: 2004
ISSN: 0176-4276,1432-0940
DOI: 10.1007/s00365-002-0531-0