On Exponential Representations of Analytic Functions in the upper Half-Plane with Positive Imaginary Part

摘要

The work presented in this paper has its origin in certain theorems which were needed by the authors for their forthcoming paper on "Finite dimensional perturbations of spectral problems and approximation methods". These theorems were closely related to results obtained by S. Verblunsky [18, 19] and H. Delange [4] but could not be deduced from those results. Thus we were led to investigate in general the question of the exponential repre¬sentation of functions analytic in the upper half-plane with positive imaginary part there, a class denoted in this paper by P. The results of that investi¬gation form the content of the paper, and the theorems which we required for our future paper as well as the theorems of Verblunsky and Delange occur as very special cases of those results.