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Title:Efficient quotient representation of meromorphic functions
Author(s):Hopkins, Kevin Walter
Doctoral Committee Chair(s):Kaufman, Robert
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Abstract:Representations of meromorphic functions as quotients of analytic functions have been studied for years. Miles showed that any meromorphic function f can be written as f$\sb1$/f$\sb2$ where each f$\sb{\rm j}$ is entire and T(r,f$\sb{\rm j}$) $\leq$ AT(Br,f). This result is trivial if the pole set Z is finite. For an infinite pole set Z of f, he established the existence of entire f$\sb{\rm j}$ such that T(r,f$\sb{\rm j}$) $\leq$ A$\sp\prime$N(B$\sp\prime$r,Z).
Miles's technique, called balancing, was to add elements to the pole set Z in such a way that he could apply a result of Rubel and Taylor. Our technique is to add elements Z$\sp\prime$ and Z$\sp{\prime\prime}$ to the pole set Z in such a manner as to make the Fourier coefficients of $\rm\log\ \vert f\sb2(re\sp{i\theta})\vert$ small. We need to ensure that the number of zeros added in the balancing does not make N(r, Z $\cup$ Z$\sp\prime$ $\cup$ Z$\sp{\prime\prime}$) $\gg$ N(r,Z). We also need to ensure that the zeros added to make one coefficient small do not adversely interact with other coefficients.
Miles exhibited a function where T(r,f$\sb{\rm j}$) $\leq$ AT(r,f) is not possible on some sequence of r's. In this thesis we examine the cases where we can set the constant B to equal one in Miles's result.
We are able to achieve A = 1 + o(1) and B = 1 on a sequence of r$\sb{\rm n}$'s. For a meromorphic function f of finite order we achieve$$\rm A = O(\rho\ \max\ \left\{1,{n(r,Z)\over N(r,Z)}\right\})$$with B = 1 on a set of r's of positive logarithmic density. For a meromorphic function f of infinite order we obtain a more complicated expression for A with B = 1 on a set of positive logarithmic density.
Issue Date:1989
Rights Information:Copyright 1989 Hopkins, Kevin Walter
Date Available in IDEALS:2011-05-07
Identifier in Online Catalog:AAI8924841
OCLC Identifier:(UMI)AAI8924841

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