Operators with discontinuous range and their applications

The paper contains a brief review of applications of operators with discontinuous range (called discontinuous operators) to problems of the approximation theory and ill-posed problems. In the approximation theory, a class of integral operators with delta-type kernels is well known. Applying these operators to continuous functions defined on an interval, one can obtain uniform approximations of these functions only inside the interval. To obtain uniform approximations on the whole interval, we construct discontinuous operators using operators with delta-type kernels. The general idea of this construction proposed by A. P. Khromov was used by the author for solving some problems: approximation of derivatives of any order, reconstruction of continuous functions, reconstruction of derivatives of a function on an interval by given root-mean-square approximations of the function. Finally, for Abel integral equations with approximate right-hand sides, we develop a simple regularization method, which does not require any additional information on the exact solution except for its continuity. The problems of the choice of a regularization parameter and the error estimate are solved in the case where the exact solution of Abel equation belongs to a Lipschitz class. Moreover, a new class of discontinuous operators with polynomial finitely supported kernels is introduced and applications of these operators in the problems of approximation and reconstruction functions and their derivatives are discussed.