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

Abstract

We prove a Hadamard-type theorem which connects the generalized order of growth $\rho^*_f(\alpha, \beta)$ of entire transcendental function $f$ with coefficients of its expansion into the Faber series. The theorem is an original extension of a certain result by S. K. Balashov to the case of finite simply connected domain $G$ with the boundary $\gamma$ belonging to the S. Ya. Al'per class $\Lambda^*.$ This enables us to obtain boundary equalities that connect $\rho^*_f(\alpha, \beta)$ with the sequence of the best polynomial approximations of $f$ in some Banach spaces of functions analytic in $G$.