Finite sections for singular integral operators by weighted Chebyshev polynomials
type: Item
instance of: scholarly article
title: Finite sections for singular integral operators by weighted Chebyshev polynomials
zbMATH Open document ID: 0829.45012
DOI: 10.1007/BF01299966
author: Peter Junghanns
author: Steffen Roch
author: U. Weber
published in: Integral Equations and Operator Theory
publication date: 1996-01-24T00:00:00Z
review text: The authors consider the equations (1) \(au + bS_\Gamma u = f\), \(au + S_\Gamma bu = f\) with piecewise continuous coefficients \(a,b\) on the interval \(\Gamma = [-1,1]\), with \(f \in L_2 (\Gamma)\) and with the Cauchy singular integral operator \(S_\Gamma \varphi (x) = \int_\Gamma (y - x)^{-1} \varphi (y)dy\). Equations (1) are replaced by the discrete analogues (2) \(P^\nu_n (aI + bS_\Gamma) P^\nu_nu = P^\nu_nf\), \(P^nu_n(aI + S_\Gamma b) P^\nu_nu = P^\nu_nf\), where \(I\) is the identity and \(P_n \{\sum^\infty_0 \xi_k u^\nu_k\} : = \sum^{n - 1}_0 \xi_k u^\nu_k\) are projectors on the \(n\)-dimensional subspaces generated by weighted trigonometric polynomials \(\{u^\nu_k\}^\infty_0\), \(\nu = 1,2,3,4\). The authors find conditions for the \(L_2\)-convergence of the solutions of the approximating equations (2) to the solution of the original equation (1) (i.e. the stability conditions). To formulate the main result of the paper let us recall the following notations from the paper: Denote by \(T = \{\zeta: |\zeta |= 1\}\) the unit circle then \(a^{(R)} (\zeta) : = a(Re \zeta)\) is a special lifting of a function \(a(x)\) from \(\Gamma\) to \(T\). Theorem: For piecewise-continuous coefficients \(a\) and \(b\) the finite section method (2) is stable if and only if the following holds: (i) \(A\) is an invertible operator; (ii) \([a^{(R)} I - b^{(R)}s] (I + S_T) + (I - S_T)\) is an invertible operator in \(L_2 (T)\); (iii) for every \(x \in (-1,1)\) the origin lies outside the triangle with vertices at the points \([a(x + 0) - b(x + 0)]/[a(x - 0) - b(x - 0)]\), \([a(x + 0) + b(x + 0)]/[a(x - 0) + b(x - 0)]\), and 1.
Mathematics Subject Classification ID: 45P05
Mathematics Subject Classification ID: 45E05
Mathematics Subject Classification ID: 45L05
Mathematics Subject Classification ID: 47G10
zbMATH DE Number: 765286
zbMATH Keywords: weighted Chebyshev polynomials
zbMATH Keywords: convergence
zbMATH Keywords: Cauchy singular integral operator
zbMATH Keywords: stability
zbMATH Keywords: finite section method
reviewed by: Roland Duduchava
MaRDI profile type: Publication
cites work: Q3866607
cites work: Numerical analysis for integral and related operator equations.
cites work: Q3997999
cites work: Q5748468
full work available at URL: https://doi.org/10.1007/bf01299966
OpenAlex ID: W2003106611
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description: scientific article; zbMATH DE number 765286