The Schur Transformation for Generalized Caratheodory functions
by
Ekaterina Lopushanskaya
Voronezh State University
Coauthors: Mihaly Bakonyi (Georgia State University)

We define the Schur algorithm for generalized Caratheodory functions and study its properties.

The function f(z) is called a generalized Caratheodory function with k negative squares if it is meromorphic in the open unit disc D and the kernel

Kf(z, w)= f(z)+f(w)* 1-zw*

has k negative squares in the domain of holomorphy of f(z) in D. We denote this class of functions by C_k.

Let f ∈ C_k^z₁ has the Taylor expansion

f(z)= ∞ å i=0  ci(z-z1)i

and let f1(z) be the Schur transformation of f(z).Then f1 ∈ Cz1k1, where

if Re c₀ ≠ 0 and c₀+c₀^* > 0, then k₁=k

if Re c₀ ≠ 0 and c₀+c₀^* < 0, then k₁=k-1

if Re c₀=0, then k₁=k-k, where k > 0 is the smallest integer such that c_k ≠ 0.

The research is supported by the Russian Foundation for Basic Research, grant RFBR 08-01-00566-a

Date received: May 18, 2008