An optimal spline-method for solving operator equations
type: Item
instance of: scholarly article
title: An optimal spline-method for solving operator equations
zbMATH Open document ID: 1045.65046
author: B. G. Gabdulkhaev
author: I. K. Rakhimov
published in: Russian Mathematics
publication date: 2003-08-06T00:00:00Z
review text: The authors consider a spline method for approximately solving operator equations of the form \(Gx+Tx=y\), where \(G\) and \(T\) are linear operators between spaces \(X\) and \(Y\) which are any of \(2\pi\)-periodic function spaces \(L_2\) and \(W_2^1\), \(G\) and \(G+T\) are continuously invertible, and \(T\) is completely continuous. The approximate solution, by the Galerkin method, is sought using the pair \((X_n, Y_n)\) of spaces, where \(X_n\) is the space of all \(2\pi\)-periodic first order splines with \(2n+1\) equidistant nodes between \(-\pi\) and \(\pi\), and \(Y_n\) is a space of trigonometric polynomials. Overall assumptions in the paper are that the orthogonal projection \(\Phi_n\) from \(Y\) onto \(Y_n\) satisfies \(\Phi_n G=G\Phi_n\) on \(X_n\) and \(Gx+Tx=y\) is uniquely solvable for all \(y\in Y\). The authors establish unimprovable, in certain sense, error estimates and prove the optimality by order of the method. The applicability of the method obtained in the general setting to both regular and singular integral and integral-differential equation, is also established. Reviewer's remark: A good contribution in the area of numerical functional analysis.
reviewed by: M. Thamban Nair
Mathematics Subject Classification ID: 65J10
Mathematics Subject Classification ID: 47A50
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45B05
Mathematics Subject Classification ID: 45E10
Mathematics Subject Classification ID: 45J05
zbMATH DE Number: 1960450
zbMATH Keywords: operator equations
zbMATH Keywords: Galerkin method
zbMATH Keywords: spline method
zbMATH Keywords: trigonometric polynomials
zbMATH Keywords: singular integral equations
zbMATH Keywords: Fredholm equation
zbMATH Keywords: integral-differential equation
zbMATH Keywords: error estimates
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1960450