On integral representations in the Hardi classes and on the best approximations in some functional spaces

Abstract

For the case of a simply connected domain in the plane one proves necessary and sufficient conditions for the representation of functions of the Hardy class $H^1$ by an integral with respect to the harmonic measure of its boundary values. A theorem is given, characterizing the rate of decrease of the best polynomial approximations of an entire function in Hardy classes by the order and the type of this function.