Bases and isomorphisms in spaces of analytic functions

Official URL: http://risc01.sabanciuniv.edu/record=b1161882 (Table of Contents)

We will discuss the construction of bases in a space of analytic functions for a given domain and isomorphic classification of spaces of analytic functions. We will focus on results in one dimensional case. In one dimensional case, we consider the construction of bases in two different ways. Using one of them, we construct interpolational bases for the space of analytic functions on a compactum K and in that part, results of Leja, Walsh, and Zahariuta are used. Then, isomorphic classification follows by the use of Potential Theory. Using the second way, we construct a common basis for the spaces of analytic functions of a regular pair “compact set domain” by the Hilbert methods that was proposed by Zahariuta. GKS-duality is used for both of the cases. In multidimensional case, some results about bases and isomorphisms of spaces of analytic functions in several variables that were proved by Zahariuta are represented (see also Aytuna). Since a multidimensional analogue of GKS-duality does not exist, interpolational bases cannot be constructed as in one dimensional case. But the bases constructed by Hilbert methods proves to be applicable for studying the isomorphism of the space of analytic functions on D to the space of analytic functions on the unit circle of n-dimensional complex plane. Keywords: Hilbert scales, spaces of analytic functions, Green potential, regularity, GKS-duality.