Best Polynomial Approximation in Lp-Norm and (p,q)‎-Growth of Entire Functions

Abstract EN

The classical growth has been characterized in terms of approximation errors for a continuous function on [-1,1] by Reddy (1970), and a compact K of positive capacity by Nguyen (1982) and Winiarski (1970) with respect to the maximum norm.

The aim of this paper is to give the general growth ((p,q)-growth) of entire functions in ℂn by means of the best polynomial approximation in terms of Lp-norm, with respect to the set Ωr={z∈Cn; expVK(z)≤r}, where VK=sup{(1/d)log|Pd|,Pd polynomial of degree ≤d, ∥Pd∥K≤1} is the Siciak's extremal function on an L-regular nonpluripolar compact K is not pluripolar.