Zheer S. The best polynomial approximation of entire transcendential functions of generalized order in complex area

A research object are whole transcendent functions of one variable and generalized descriptions of their growth. The purpose of work consists in research of conduct of the best polynomial approximations of whole transcendent functions depending on such their descriptions as the generalized orders of growth. The modern methods of structural theory of functions are in-process utillized actual and complex variables, functional analysis, theory of analytical functions, in particular methods researches, developed in works of S. N. Bernshteyn, A. V. Batyrev, S. B. Vakarchuk et al. Dissertation work is devoted to research of questions of the best polynomial approximations of the entire transcendential functions in banach spaces, viz were established correlations, which determine generalized orders of growth of the entire transcendential functions through their best polynomial approximation along the contour and on area. Were received limiting correlations which link between themselves various generalized characteristics of growth of entire transcendential functions f and their best polynomial approximations in integral metrics of banach spaces Hardy, Bergman or analytic in the circle of the unit radius functions. Were proved theorems of Adamard type which link the generalized orders of growth of the whole transcendental functions f with factors of their decomposition in rows of Faber in final one-coherent area G. These theorems are distribution of results of S. K. Balashov and M. N. Sheremeta from a circle of unit radius on one-coherent area of a complex plane. On the basis of this were received limiting equalities which link among themselves generalized orders of the whole transcendental functions f and sequence of their best polynomial approximations in banach spaces. Job performances and methods of their receipt can be applied in subsequent researches of structural theory of functions of complex variable, functions related to polynomial approximations.