Approximation of solutions to singular integro-differential equations by Hermite–Fejer polynomials

Singular integral and integro-differential equations have a lot of applications and thus were thoroughly studied by domestic and foreign mathematicians since the beginning of 20th century, and by the 70th years the theory of such equations was finally completed. It is known from this theory that the exact solutions to such equations exist only in rarely particular cases, so since that time the approximate methods for solving these equations as well as the techniques of the justification of these methods were developed. Justification of the approximate method means the proof of the existence and the uniqueness of the approximate solution, estimation of its error and the proof of the convergence of the approximate solutions to the exact solution. Moreover, to compare the approximate methods in different aspects, the optimization theory for approximate methods was created.
However, sometimes, depending on the particular problem, an important role is also played by the form of an approximate solution. For instance, sometimes it is desirable to have an approximate solution as a spline, sometimes, as a polynomial, sometimes it is enough to have just the approximate values of the solution at the nodes. It is quite obvious that depending on the kind of the approximate solution the technique of the justification of the method should be chosen. Unfortunately, there are very few of such techniques, that is why the theory of justification of the approximate methods is now intensively studied.
In the present work we justify an approximate method for solving singular integro-differential equations in the periodic case. An approximate solution is sought as a trigonometric interpolation Hermite-Fejer polynomials. For justification of this approximate method, the technique developed by B.G. Gabdulkhaev and his pupils is used. The convergence of the method is proved and the errors of the approximate solutions are estimated.