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

Abstract

JOZEF SICIAK1. Introduction.Let £ be a bounded closed set in the space C" of n-complex variables z = (zx, ...,zB).Let fe(z) be a real function defined and bounded on E. In the following we define an extremal function <&(z, E, fe), z e C, depending on £ and fe.For this purpose we introduce a triangular array of extremal points {?iv)} 0I •£• In tne case tnat b(z) is lower semicontinuous, the formal definition of the points ykv) is analogous to the definition of Fekete-Leja's point of a plane set.In the case that E is in C1 and fe(z) = 0, the points y^v) are exactly Fekete's points of £ (see (5.3')).In the case of one complex variable, the function log 0(z, £, 0) is a generalized Green's function for the unbounded component of CE with pole at co.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 í>(z,£,0), zeC", 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 |Pv(z)| ^ M<Dv(z,E,0), zeC", M = max |Py(z)\ PjLz) being an arbitrary polynomial of order v, v = 0,1,_This inequality is useful in the proof of the following theorem: // í>(z,£,0) is continuous in C" and ER is given by ER = {z|O(z,£,0)<P} , P>1, then the necessary and sufficient condition that function f(z) be holomorphic in ER and not continuable to holomorphic (single-valued) function in any ER., R' > R, is that (*) Hm sup 7 max |/(z) -ttv(z) | = -, v->a> ieE "where nv(z) denotes a polynomial of order v of the Tchebycheff best approximation tof(z) on E.