On the best polynomial approximation of entire transcendental functions in Banach spaces. II

Abstract

We study the behavior of the best approximationsE n (ϕ) p of entire transcendental functions ϕ(z) of the order ρ=∞ by polynomials of at mostn th degree in the metric of the Banach space E′p(Ω) of functions /tf(z) analytic in a bounded simply connected domain Ω with rectifiable Jordan boundary and such that $$\left\| f \right\|_{E'_p } = \left\{ {\iint_\Omega {\left| {f\left( z \right)} \right|^p }dxdy} \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}}< \infty $$ . In particular, we describe the relationship between the best approximationsE n (ϕ)p and theq-order andq-type of the function ϕ(z).