Effectiveness of similar sets of polynomials of two complex variables in polycylinders and in Faber regions
The authors consider a basic set \(\{P_{m,n} (z_1,z_2)\}\) of polynomials in the two complex variables \(z_1\) and \(z_2\), in two different regions. The first region is a polycylinder and the second region is a Faber region. When the set is a base then every polynomial of two complex variables is a linear combination of elements from this set. Hence \(z_1^m z_2^n\) will be uniquely represented as a finite sum of polynomials from the given set. This in turn makes it possible to represent every function of two complex variables, which is regular in a neighborhood of the origin, as an infinite sum of polynomials from the set. Definitions of Cannon functions and Cannon sum are given. The authors use functional analysis to prove few inequalities dealing with the bounds of the Cannon functions in both the polycylinder and the Faber regions.
scientific article; zbMATH DE number 1180650
scientific article