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Description:
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 $S_r \subset \R^N$.