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

Abstract

The classical growth has been characterized in terms of approximation errors for a continuous function on $[-\mathrm{1,1}]$ 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 ${ℂ}^n$ by means of the best polynomial approximation in terms of $L^p$-norm, with respect to the set ${\mathrm{\Omega }}_r=\{z\in {\mathbf{C}}^n;\exp V_K(z)\le r\}$, where $V_K=\sup \{\mathrm{(1/}\mathrm{d)}\log |P_d|,P_d\text{polynomial}\text{of}\text{degree}\le d,\parallel P_d{\parallel }_K\le 1\}$ is the Siciak's extremal function on an $L$-regular nonpluripolar compact $K$ is not pluripolar.