On the best polynomial approximation of entire transcendental functions of generalized order

We prove a Hadamard-type theorem that associates the generalized order of growth of an entire transcendental function ƒ with the coefficients of its expansion in a Faber series. This theorem is an extension of one result of Balashov to the case of a finite simply connected domain G with boundary γ belonging to the Al'per class Λ*. Using this theorem, we obtain limit equalities that associate with a sequence of the best polynomial approximations of ƒ in certain Banach spaces of functions analytic in G. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 8, pp. 1011–1026, August, 2008.