�ber eine in gewissen Bereichen mit Maximumfl�che g�ltige Integraldarstellung der Funktionen zweier komplexer Variabler. I

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

“…Theorem 1. // the complement of E is connected, O(z) = í>(z, E,0) is continuous in C" and the function fiz) is holomorphic on E, then the sequence of interpolating polynomials (1) Lv(z,/) = I f(yy->) L(i)(z,)>w), v = 1,2 ..., i= i…”

“…We shall now find the absolute value of the determinant (2.4) which corresponds to the system p(v) of points (1 where Pv_ki(z2,...,z") is a polynomial of degree v -fct of the n -1 variables z2, ...,z". To begin with, let us assume that do = 0.…”

“…Lagrange interpolation formulae for polynomials of n complex variables(2). Let Py(z) = Pv(zx, ...,z") be a polynomial of degree v (1) P,(z) = I aki...knz^...zk".…”

“…It follows from (8) that if Eko for some 1 ^ fc0 g n is not regular, then í>(z,£,0), £ = El x ... x £", is discontinuous at some points outside of £, e.g., if í>(zi,£i,0) is not continuous at a point z°eEu then the function O(z,£,0) is discontinuous at any point iz°,z2,...,z"), where (z2,...,z") ^£2 x ... x £". Given any p > 1 we define Ep and and Cp by (1) £p = {z|<D(z)<p}, Cp = {z|<D(z) = p}.…”

Functions of extended class in the theory of functions of several complex variables

“…In introducing this extended class, we make the following requirements: (1) In the case of a bicylinder, the extended class coincides with doubly harmonic functions. (2) In the case of a domain which by a pseudo-conformal transformation can be obtained from a bicylinder, the class coincides with the functions obtained by a pseudo-conformal transformation from doubly harmonic functions.…”