Markov – Stieltjes transformation of measures and some of its applications

Abstract

The paper is devoted to the study of the properties of the Markov – Stieltjes transformation of measures. In the works of J. Anderson, A. A. Pekarsky, N. S. Vyacheslavov, E. P. Mochalina et al., the functions of Markov – Stieltjes type were studied from the point of view of the approximation theory. In the works of A.R. Mirotin and the author, the Markov – Stieltjes transform of functions was studied as an operator in Hardy and Lebesgue spaces. In this paper, the general properties of the Markov – Stieltjes transform of measures are studied, the theorem of analyticity and the uniqueness theorem are proved, the Markov – Stieltjes transformations of positive and complex measures are described, the inversion formula and the continuity theorem are established, the boundary behavior of the given transformation is investigated. In particular, the analogues of the Sokhotsky – Plemelya formulas are established. Applications to the theory of self-conjugate operators are given. In addition, the results obtained can find use in the theory of functions and integral operators, as well as in the theory of information transfer, in particular, in the theory of signal processing.