Bernstein-Walsh type theorems for real analytic functions in several variables

Abstract

The aim of this paper is to extend the classical maximal convergence theory of Bernstein and Walsh for holomorphic functions in the complex plane to real analytic functions in R^N. In particular, we investigate the polynomial approximation behavior for functions $F: L to C, L= (Re z, Im z ) : z in K$, of the type $F= g overline h$, where g and h are holomorphic in a neighborhood of a compact set $K subset C^N$. To this end the maximal convergence number $rho(S_c,f)$ for continuous functions f defined on a compact set $S_c subset C^N$ is connected to a maximal convergence number $rho(S_r,F)$ for continuous functions F defined on a compact set $S_r subset R^N$.