Vol. 25, No. 1, 1968

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A representation of a bounded function as infinite product in a domain with Bergman-Shilov boundary surface

Maciej Skwarczyński

Vol. 25 (1968), No. 1, 177–183

In this paper we consider a closed domain Q in the space of two complex variables along with its decomposition into two parameter family q(λ12) of segments of analytic surfaces. Under some additional assumptions about the domain Q one introduces using Poisson formula the real-valued functions of the real extended class. These functions are harmonic in each q(λ12). This in turn enables us to define the complex valued functions of the complex extended class. The aim of the paper is to show that a bounded analytic function which has infinitely many zero surfaces ns(z) = 0, z = (z1,z2), s = 1,2, can be represented in the domain Q in the form

f(p) = w(p)   ek(p),p ∈ Q, w(p) ⁄= 0.

Here ek(p) are functions of the complex extended class.

Mathematical Subject Classification
Primary: 32.02
Received: 23 March 1967
Published: 1 April 1968
Maciej Skwarczyński