Best L_p-approximation of analytic functions and generalized (alpha, alpha)-growth

Abstract

The aim of this paper is the characterization of the generalized order and generalized type of analytic functions of several complex variables in the open set

Ω_R = {z ∈ ℂⁿ; exp V_E(z) < R }

in terms of the best polynomial approximation on a compact set E in L_p-norm and the coefficients of the development with respect to the extremal polynomial according to the set

Ω_r = {z ∈ ℂⁿ; exp V_E(z) ≤ r}, 1 < r < R,

where

V_E = sup { (1 ∕ d) ln | P_d|, P_d a polynomial of degree ≤ d, |P_d|_E ≤ 1}

is the Siciak extremal function of a L-regular compact E which defined the pluri-complex Green function with a pole at infinity.