ИСТИНА

The project is aimed at solving fundamental problems of the theory of functions of a real and complex variable and related problems of harmonic analysis. This is the first time a research team has been formed in its current composition. Modern tasks are set at the junction of the directions studied by different members of the team. These are promising fundamental problems related to the study of trigonometric series, sine and cosine series, expansions in the trigonometric system, analytical functions. During the implementation of the project, an important question in the theory of functions about the asymptotics near zero of cosine series with monotone coefficients will be investigated. It is supposed to obtain the final necessary and sufficient conditions in terms of fractional monotonicity for this asymptotic behavior. The next task of the project is to describe the set of zeros on the unit circle of a power series with an absolutely convergent series of coefficients. A study of the behavior in a neighborhood of zero of sums of converging trigonometric series will be carried out. It is also planned to investigate sine series with convex varying coefficients of the coefficients and to obtain exact two-sided estimates of the sums for this class, which strengthen the classical results. In the theory of analytic functions, new lower bounds for the minimum of the modulus on the circles of analytic and entire functions will be obtained. A promising detailed study will be carried out (so far not done by any of the specialists in the theory of entire functions), which will give an answer to the question: what, in fact, can give meaningful and new ideas of M.A. Evgrafova. In the theory of multiple trigonometric series, the question of the absolute convergence of a series of moduli of the coefficients of the expansion of a function in terms of the trigonometric system in terms of the modulus of continuity of the function will be investigated. This topic is in line with actively developing research both in our country and abroad, and is widely represented at various international conferences on the theory of functions.