The estimation of the error resulting due to the truncation applied to homogeneous integral equations with Cauchy's kernel (Q1879580)

Consider the following Sturm-Liouville problem: \[ {\tfrac 1 \pi}\;\int_{-c}^c\;{\frac {\varphi_- (t,\gamma)\;dt}{1-e^{i(x-t)}} } \;=\;\sum_{n=1}^\infty {\tfrac 1 n}\;Q_n(\gamma)\;\Phi_{n-}(\gamma)\;( e^{inx} + e^{-inx})\;,\;Q_n(\gamma)\;=\;o({\tfrac 1 n})\;; \] and an approximate operator: \[ {\tilde K}\varphi_-(\gamma)\;=\;\varphi_- (x,\gamma) \] \[ \;-\;{\frac {R(x)}{2\pi^2}}\;\int_{-c}^c\;{\frac {e^{it}\;dt}{R(t)(e^{it} - e^{ix})} } \int_{-c}^c\;\sum_{n=2}^\infty {\frac {Q_n(\gamma)} n}\;(e^{in(t-y)} + e^{-in(t-y)}) \varphi_- (y,\gamma)\;dy . \] Using a theorem of \textit{Yu. I. Chersky} [Dokl. Akad. Nauk SSSR 150, 271--274 (1963; Zbl 0183.11902)] and the discrete Fourier transform, the author proves that the inverse operator \({\tilde K}^{-1}\) is bounded , and the Sturm-Liouville problem has a unique solution in \(L_2[-c,c] .\) The author gives also a computational example.