Best Approximation in Lp-norm and Generalized (Alpha; Beta)-growth of Analytic Functions

Abstract

Let p be a positive number or infinity, and consider a set denoted as Omega_R, which consists of all complex n-dimensional points where the exponential of a function V_E at that point is less than R, for some value of R greater than 1. The function V_E is defined as the supremum of the expression involving the logarithm of the absolute value of certain polynomials P_d, where these polynomials have a degree at most d and are bounded by 1 on the compact set E. This function V_E is known as the Siciak extremal function associated with a compact set E that satisfies a certain regularity condition.

The purpose of this paper is to describe the generalized growth of analytic functions of several complex variables within the given open set. This is done by examining the best polynomial approximation in the L_p norm on the compact set E, with respect to a related set Omega_r, which consists of all points where the exponential of V_E is at most r, for values of r between 1 and R.