On some extremal functions and their applications in the theory of analytic functions of several complex variables

Abstract

In the case of one complex variable, the function log FD(z, E, 0) is a generalized Green's function for the unbounded component of CE with pole at so. It is well known that the Green's function plays the primary role in the theory of interpolation and approximation of holomorphic functions of one variable by polynomials (see [27]). It turns out that the function D(z, E, 0), z cn Cn, also plays a quite similar role in the theory of interpolation and approximation of holomorphic functions of several variables by polynomials. For instance, one can obtain the Bernstein-Walsh inequality