Approximation of a function given by its Laurent series
type: Item
instance of: scholarly article
title: Approximation of a function given by its Laurent series
zbMATH Open document ID: 0846.41019
DOI: 10.1007/BF02142506
author: Jeannette Van Iseghem
author: Peter R. Graves-Morris
published in: Numerical Algorithms
publication date: 1996-09-10T00:00:00Z
review text: Let \(f(z)\) be a function holomorphic in an annulus \(\{z: r<|z|< \mathbb{R}\}\), given by its Laurent series, and \(p\), \(p'\), \(q\) nonnegative integers. Following the spirit of the weak Padé approximation, for a fixed integer \(n\) a rational approximating function \[ [ n,p', p,q ](x)= {{N(z)} \over {Q(z)}}, \qquad N(z)= \sum^p_{k= -p'} a_k z^k, \qquad Q(z)= \sum^q_{k=0} b_k z^k, \] is constructed in such a way that the Laurent expansion of \((Qf- N) (z)\) has no terms from power \(-n- p'\) to power \(p+ q-n\). Several good qualitative properties of these approximants are proved. Thus, in the case when \(f\) is meromorphic in a larger annulus, an analogue of the Montessus de Mallore theorem is established (for \(p, p'\to \infty\)), along with relations similar to the relations of the QD algorithm. Moreover, the denominators \(Q(z)\) are linked to reverse vector orthogonal polynomials, and recurrence relations for both \(N\) and \(Q\) are presented.
Mathematics Subject Classification ID: 41A21
zbMATH DE Number: 871292
zbMATH Keywords: Padé approximation
zbMATH Keywords: vector orthogonal polynomials
reviewed by: Andrei Martínez Finkelshtein
MaRDI profile type: Publication
cites work: Padé-type approximation and general orthogonal polynomials
cites work: Q4764025
cites work: A review of Padé methods for the acceleration of convergence of a sequence of vectors
cites work: Q5653525
cites work: Moment Theory, Orthogonal Polynomials, Quadrature, and Continued Fractions Associated with the unit Circle
full work available at URL: https://doi.org/10.1007/bf02142506
OpenAlex ID: W2080044448
description: scientific article; zbMATH DE number 871292