Continued fractions and certain functional equations
type: Item
instance of: scholarly article
title: Continued fractions and certain functional equations
zbMATH Open document ID: 0622.65137
author: B. M. Nayar
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 1986-01-01T00:00:00Z
review text: The paper deals with an application of continued fractions to a functional equation in a Hilbert space equipped with Dirichlet norm. The current effort is an extension of the work of Stesin (1983) who shows that the successive denominators of the S-fraction expansion corresponding to the Taylor series approximate the Fredholm denominator of the kernel. The author extends the above theory to the kernel of the integral equation associated with the Robin-Poincaré problem. This kernel is assumed to be a member of the symmetrizable class of kernels. It is shown that the Neumann series solution of the Fredholm-Poincaré integral equation with a kernel fully symmetrizable by a positive operator gives a Taylor series with Schwarz's constants as coefficients. The paper also contains results relating coefficients associated with Lanczos' minimized iteration scheme and those of J-fraction and S- fraction expansions. Notions of bounds, convergence and truncation errors are discussed using equivalence transformations. The meromorphic character of the solution is established using the properties of continued fractions. This provides an alternate to the Fredholm theory which is generally used for this purpose.
reviewed by: N.A.Warsi
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 65J10
Mathematics Subject Classification ID: 45B05
zbMATH DE Number: 4009763
zbMATH Keywords: Lanczos method
zbMATH Keywords: continued fractions
zbMATH Keywords: Hilbert space
zbMATH Keywords: S-fraction expansion
zbMATH Keywords: Taylor series
zbMATH Keywords: Robin-Poincaré problem
zbMATH Keywords: Neumann series
zbMATH Keywords: Fredholm- Poincaré integral equation
zbMATH Keywords: bounds
zbMATH Keywords: convergence
zbMATH Keywords: truncation errors
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4009763