Collocation methods for systems of Cauchy singular integral equations on an interval (Q2714035)

The article deals with collocation methods which, in particular, solve Cauchy singular integral equations with piecewise continuous coefficients on an interval and are based on using Chebyshev nodes of the first and second kinds. Attention is focused on choosing Chebyshev nodes as collocation points which makes it possible to construct the matrices of the discretized equations and is especially important for approximating the solution to nonlinear Cauchy singular integral equations by a sequence of solutions to linear equations. NEWLINENEWLINENEWLINEA particular attention is paid to studying necessary and sufficient conditions for the stability of the collocation method. This is achieved by constructing an operator sequence \(\{A_n\}\) belonging to a \(C^*\)-algebra \({\mathcal A}\) which is generated by the sequences of the collocation method for the equations under consideration. These stability conditions can be formulated as follows:NEWLINENEWLINENEWLINEThere exist \(*\)-homomorphisms \(W\:{\mathcal A}\to{\mathcal L}({\mathbf L}^2_{\sigma})\), \(\widetilde W\:{\mathcal A}\to{\mathcal L}({\mathbf L}^2_{\sigma})\) and \(\eta_{\pm}\:{\mathcal A}\to {\mathcal L}(\ell^2)\) such that, in the case of Chebyshev nodes of second kind, a sequence \(\{A_n\}\) in \({\mathcal A}\) is stable if and only if the operators \(W\), \(\widetilde W\), and \(\eta_{\pm}\) are invertible.