Approximate solution of singular integral equations on the real line by the polynomial quadrature method (Q1873529)

An approximate solution of the singular integral equation with Cauchy kernel on the real line \[ a(x)\phi(x) + b(x)/{}(\pi i)\int_{R} (t-x)^{-1}\phi(t) dt + \int_{R} K(x,t)\phi(t) dt = f(x),\quad x\in R\tag{1} \] is studied. This equation is considered in the spaces of complex-valued functions \(\phi(x)\) such that \(\phi(x)\) and \((x+i)\phi(x)\) belong to \(C^{(r)}(R)\) or \(H^{(r)}_{\alpha}(R),\) \(0<\alpha\leq 1,\) \(r\geq 0\) simultaneously, under assumptions which guarantee that it has a unique solution. The exact problem \((1)\) is approximated by using the Lagrange interpolation method based on the function system \(\omega_{k}(x)=[(x-i)/{}(x+i)]^{k},\) \(k=0,\pm 1, \pm 2, \dots \) and the interpolation nodes \(x_j=-\cot [\pi j/{}(2n+1)],\) \(j=-n,\dots, n.\) An approximate solution is sought in the form of a linear combination of the functions \(\omega_{k}(x)\). The convergence rate of the proposed method is comprehensively discussed and its estimates in various norms are given.