Applications of Model Theory to Complex Analysis

Author: Stroyan, Keith Duncan

Year: 1971

Degree: Dissertation (Ph.D.)

Advisors: Bohnenblust, Henri Frederic; Luxemburg, W. A. J.

Committee Member: Unknown, Unknown

Option: Mathematics

DOI: 10.7907/69R3-SY38

Abstract

We use a nonstandard model of analysis to study two main topics in complex analysis.

UNIFORM CONTINUITY AND RATES OF GROWTH OF MEROMORPHIC FUNCTIONS is a unified nonstandard approach to several theories; the Julia-Milloux theorem and Julia exceptional functions, Yosida's class (A), normal meromorphic functions, and Gavrilov's Wp classes. All of these theories are reduced to the study of uniform continuity in an appropriate metric by means of S-continuity in the nonstandard model (which was introduced by A. Robinson).

The connection with the classical Picard theorem is made through a generalization of a result of A. Robinson on S-continuous *-holomorphic functions.

S-continuity offers considerable simplifications over the standard sequential approach and permits a new characterization of these growth requirements.

BOUNDED ANALYTIC FUNCTIONS AS THE DUAL OF A BANACH SPACE is a nonstandard approach to the pre-dual Banach space for H∞(D) which was introduced by Rubel and Shields.

A new characterization of the pre-dual by means of the nonstandard hull of a space of contour integrals infinitesimally near the boundary of an arbitrary region is given.

A new characterization of the strict topology is given in terms of the infinitesimal relation: "h b k provided ||h-k|| is finite and h(z) ≈ k(z) for z∈(*D)".

A new proof of the noncoincidence of the strict and Mackey topologies is given in the case of a smooth finitely connected region. The idea of the proof is that the infinitesimal relation: "h γ k provided ||h-k|| is finite and h(z) ≈ k(z) on nearly all of the boundary", gives rise to a compatible topology finer than the strict topology.

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