On approximation methods for the solution of one dimensional singular integral equations

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Abstract
In the present paper we give a survey on some new result obtained in the last few years in the theory of approximation methods for one dimensional singular equations (among others to the methods of collocation and of mechanical quadratures). Conditions for convergence and stability of these methods in the spaces L p (1<p<∞) and H α (=Lipα)(0<α<1) are formulated and their rate of convergence is determined. simultaneously here for the first time the necessitgy of the conditions for the convergence of the methods mentioned above is investigated. In the first section a general and uniform methods for the treatment of projection methods for linear operator equations in Banach spaces in presented. The Core of this methods with unbounded projectors which is closely connected with a far-reaching generalization of the stability concept of S.G. Mikhlin.