Generalized Padé-type approximation and integral representations
type: Item
instance of: scholarly article
title: Generalized Padé-type approximation and integral representations
zbMATH Open document ID: 1060.41011
DOI: 10.1023/A:1021273702443
author: Nicholas J. Daras
published in: Advances in Computational Mathematics
publication date: 2003-04-10T00:00:00Z
review text: The author replace the multidimensional Cauchy kernel by the Bergman kernel function \(K_\Omega(t,x)\) into an open bounded subset \(\Omega\) of \(\mathbb{C}^n\) and by using interpolating generalized polynomials for \(K_\Omega(z,x)\) define generalized Padé-type approximants to any \(f\) in the space \(OL^2 (\Omega)\) of all analytic functions on \(\Omega\) which are of class \(L^2\). The characteristic property of such an approximant is that its Fourier series representation with respect to an orthonormal basis for \(OL^2(\Omega)\) matches the Fourier series expansion of \(f\) as far as possible. The errors formula and the convergence problem is studied. It is showed that the generalized Padé-type approximants have integral representations which give rise to the consideration of an integral operator which maps every \(f\in OL^2 (\Omega)\) to a generalized Padé-type approximant to \(f\). The study conclude with the extension of these ideas into every functional Hilbert space \(H\) and also with the definition and properties of the generalized Padé-type approximants to a linear operator of \(H\) into itself. There are contained two examples making use of generalized Padé-type approximants and a large list of references.
Mathematics Subject Classification ID: 41A21
Mathematics Subject Classification ID: 32E30
zbMATH DE Number: 1896711
zbMATH Keywords: generalized Padé-type approximant
zbMATH Keywords: Fourier series
zbMATH Keywords: Bergmann kernel
zbMATH Keywords: integral representation
zbMATH Keywords: integral operator
zbMATH Keywords: self-summable countable orthonormal set in a Hilbert space
reviewed by: Francisco Pérez Acosta
MaRDI profile type: Publication
full work available at URL: a:1021273702443
OpenAlex ID: W2293239716
description: scientific article; zbMATH DE number 1896711