On a Special Caratheodory Subclass and the Set of Solvability of the Markov Trigonometric Moment Problem with Periodic Gaps

Abstract

The classical Markov trigonometric moment problem is considered in the case when the moment sequence has periodically repeating gaps. The suggested approach represent a development of the N. I. Akhieser's and M. G. Krein's idea on the relation between the trigonometric moment problem and the Caratheodory coefficients problem. We associate with the gaps distribution law M a certain subclass 𝒞(M) of the Caratheodory functions class 𝒞. The multiplicative representation of the class 𝒞(M) is obtained. On this basis the complete description of the solvability set of the Markov trigonometric moment problem with gaps as well as a method of searching of its canonical solutions are given.