The best polynomial approximations of entire transcendental functions with generalized growth order in Banach spaces $ \mathcal{E}_p^{\prime}(G) $ and $ {\mathcal{E}_p}(G) $ , p ⩾ 1

Abstract

A theorem of the Hadamard type for entire transcendental functions f, which have a generalized α-order of growth ρ α (f), has been obtained. This theorem connects the values $ \widetilde{M}\left( {f,r} \right)\;\left( {r > 1} \right) $ and the coefficients a n (f) $ \left( {n \in {\mathbb{Z}_{+} }} \right) $ of the expansion of f in Faber series in a finite domain D whose boundary γ belongs to the Al’per class. This result is the extension of a result obtained by M. N. Sheremeta onto a simply connected domain. The necessary and sufficient conditions for an analytic function $ f \in \mathcal{E}_p^{\prime}(G) $ or $ f \in {\mathcal{E}_p}(G)\;\left( {1 \leqslant p \leqslant \infty } \right) $ to be entire transcendental with a generalized α-order of growth ρ α (f) are obtained. These conditions include the best polynomial approximations of the function f and determine the rate of their convergence to zero, as the degree of polynomials increases.