Applications of Model Theory to Complex Analysis

Citation

Stroyan, Keith Duncan (1971) Applications of Model Theory to Complex Analysis. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/69R3-SY38. https://resolver.caltech.edu/CaltechTHESIS:04112018-095018367

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 W_p 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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