Solution of singular integral equations by the method of oscillating functions (Q1048365)

The following complete singular integral equation with Hilbert kernel is considered \[ Ax\equiv a(s)x(s)+\frac{b(s)}{2\pi} \int_{0}^{2\pi}\cot\frac{\sigma-s}{2}x(\sigma)d\sigma+ \frac{1}{2\pi}\int_{0}^{2\pi}h(s,\sigma)x(\sigma)d\sigma=y(s). \] Here \(x(s)\) is the unknown function, \(a(s), b(s), y(s)\) and \(h(s,\sigma)\) are known complex-valued \(2\pi\)-periodic functions. Moreover, \(a(s)\) and \(b(s)\) are assumed to be continuous, and \(y(s)\) and \(h(s,\sigma)\) are square summable in the \([0,2\pi]\) and \([0,2\pi]^2\) respectively. For the approximate solution of this equation the method of oscillating functions is proposed and theoretically justified. Also the author establishes the convergence rate of the method. In conclusion it is shown that the method of oscillating functions is order-optimal among all projective methods, as well as among all direct methods which allow one to construct an approximate solution in the form of polynomial \(x_n(s)=\sum_{k=-n}^{n}c_k e^{iks}\).