Carlos Diaz-Mendoza

Quadrature on the half-line and two-point Pade´ approximants to Stieltjes functions—II: convergence

Abstract Let α be a distribution function on [a,b] (0 ⩽ a c k = ʃ b a dα(x) exist for all the integers k. The main course of the paper is to give convergence results both for sequences of two-point Pade approximants to the Cauchy transform of the distribution α (Stieltjes function) and for quadrature formulas based on exactly integrating Laurent polynomials. The case when dα(x) = k(x)dx, k(x) being a possibly complex function is also considered and estimates of the rate of convergence are given.

Quadrature on the half line and two-point Pade´ approximants to Stieltjes functions: subtitle: Part III. The unbounded case

Abstract Let α be a general, absolutely continuous measure, possibly complex, supported on [0,∞). Let Fα(z) denote its Cauchy transform. In this paper we prove, under suitable conditions, the convergence of two-point Pade type approximants to Fα and of the associated quadrature formulas ∑ j=1 n A j ƒ(x j ) to the intergral ∫ 0 ∞ f(x) d α(x) . These quadrature formulas can be of Gaussian or of interpolatory type. Estimates for the rate of convergence are also included.

Orthogonality and recurrence for ordered Laurent polynomial sequences

We consider orderings of nested subspaces of the space of Laurent polynomials on the real line, more general than the balanced orderings associated with the ordered bases {1,z^-^1,z,z^-^2,z^2,...} and {1,z,z^-^1,z^2,z^-^2,...}. We show that with such orderings the sequence of orthonormal Laurent polynomials determined by a positive linear functional satisfies a three-term recurrence relation. Reciprocally, we show that with such orderings a sequence of Laurent polynomials which satisfies a recurrence relation of this form is orthonormal with respect to a certain positive functional.

Orthogonal Laurent polynomials corresponding to certain strong Stieltjes distributions with applications to numerical quadratures

In this paper we shall be mainly concerned with sequences of orthogonal Laurent polynomials associated with a class of strong Stieltjes distributions introduced by A.S. Ranga. Algebraic properties of certain quadratures formulae exactly integrating Laurent polynomials along with an application to estimate weighted integrals on [-1,1] with nearby singularities are given. Finally, numerical examples involving interpolatory rules whose nodes are zeros of orthogonal Laurent polynomials are also presented.

Two-point Padé-type approximation to the cCuchy transform of certain strong distributions

Abstract In this paper we are mainly concerned with the Cauchy transform of certain strong distributions satisfying a type of symmetric property introduced by A.S. Ranga. Algebraic properties of the corresponding two-point Pade-type approximants are given along with results about convergence for sequences of such approximants

Padé approximants and quadratures related to certain strong distributions

Letbe a c-inversive strong distribution as de5ned in (Ranga, Numer. Math. 68 (1994) 283). In this paper, two-point Pad#e approximants and quadrature formulas related to the distributionare analyzed. c 2001 Elsevier Science B.V. All rights reserved. MSC: 41A21; 30E05; 34E05