R. Orive

Quadrature on the half-line and two-point Pade´ approximants to Stieltjes functions. Part I: algebraic aspects

Abstract Let f(z) be a Stieltjes function with asymptotic expansions L0 and L∞ at z = 0 and z = ∞, respectively. Let [ k n ] denote the two-point Pade approximant of type (m,n) which matches k of the coefficients of the series L0 and which takes its remaining interpolation conditions from L∞. We discuss in this paper the algebraic aspects of this problem. We shall emphasize the relation between quadrature formulas and two-point Pade approximants and derive expressions for the error. In a subsequent paper we shall consider the convergence aspects of these approximants. For example, the positivity of the error, which is obtained here, will result in the monotonic convergence of certain sequences of two-point Pade approximants, a property which is well known in the case of classical Pade approximants.

On asymptotic zero distribution of Laguerre and generalized Bessel polynomials with varying parameters

Abstract We study the asymptotic zero distribution of Laguerre L n ( α n ) and generalized Bessel B n ( α n ) polynomials with the parameter α n varying in such a way that the limit of α n / n exists. Our approach is based on a non-hermitian orthogonality satisfied by these sequences of polynomials. In the cases that remain open we formulate the corresponding conjectures.

Riemann--Hilbert analysis for Jacobi polynomials orthogonal on a single contour

Classical Jacobi polynomials Pn(α, β), with α,β > - 1, have a number of well-known properties, in particular the location of their zeros in the open interval (-1, 1). This property is no longer valid for other values of the parameters; in general, zeros are complex. In this paper we study the strong asymptotics of Jacobi polynomials where the real parameters αn, βn depend on n in such a way that limn→∞ αn/n = A, limn→∞ βn/n = B with A, B ∈ R. We restrict our attention to the case where the limits A, B are not both positive and take values outside of the triangle bounded by the straight lines A = 0, B = 0 and A + B + 2 = 0. As a corollary, we show that in the limit the zeros distribute along certain curves that constitute trajectories of a quadratic differential.The non-hermitian orthogonality relations for Jacobi polynomials with varying parameters lie in the core of our approach; in the cases we consider, these relations hold on a single contour of the complex plane. The asymptotic analysis is performed using the Deift-Zhou steepest descent method based on the Riemann-Hilbert reformulation of Jacobi polynomials.

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.

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

Optimization of two-point Pade´-type approximants

Abstract In this paper, we first give characterization theorems for the best two-point Pade-type approximants (2PTAs) in the uniform norm. Secondly, we consider sequences of 2PTAs in a domain of the complex plane from the viewpoint of the asymptotic degree of convergence, and we also give conditions for geometric convergence.

On the convergence of bivariate two-point Padé-type approximants

Some choices of denominators are given which ensure the geometrical convergence of certain convergence of bivariate two-point Pade-type approximants to functions being holomorphic on certain domains

On a family of multivariate two-point rational approximants

Abstract The concept of two-point Pade-type approximant is extended to functions of two complex variables by means of a linear functional acting on the space of bivariate Laurent polynomials. Further, the two-point Pade approximants are defined as higher-order 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