Mathematics - Dissertations

Title

On the preservation of certain properties of functions under composition

1994

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Mathematics

Daniel Waterman

Keywords

homeomorphisms, convex functions, fourier series

Subject Categories

Number Theory | Partial Differential Equations | Physical Sciences and Mathematics

Abstract

Let $I\sb{n,m},\ m = 1,2,\...,k\sb{n}$ be disjoint closed intervals such that for each $n,\ I\sb{n,m-1}$ is to the left of $I\sb{n,m}.$ Given x, if for every $\epsilon>0$ there exists N such that $I\sb{n,m}\subset(x,x + \epsilon)$ whenever $n>N,$ then ${\cal I} = \{I\sb{n,m}:n = 1,2,\...;\ m = 1,2,\...,k\sb{n}\}$ is called a right system of intervals (at x). A left system is defined similarly. Let$$\alpha\sb{n}({\cal I}) = \sum\sbsp{i=1}{k\sb{n}}{f(I\sb{n,i})\over i}\quad{\rm where}\quad f(\lbrack a,b\rbrack) = f(b) - f(a).$$In Chapter 1 we prove the following result:

Theorem 1. If f is regulated, then $f\ \circ\ g$ has everywhere convergent Fourier series for every homeomorphism g is f and only if $\lim\limits\sb{n\to\infty}\alpha\sb{n}({\cal I}) = 0$ for every system ${\cal I}$ and for every x.

Goffman and Waterman proved an analogous theorem for the case where f is continuous.

In Chapter 2 we turn our attention to functions of bounded. $\Lambda$-variation, which we define as follows: Suppose $\Lambda = \{\lambda\sb{n}\}$ is an increasing sequence such that $\sum\limits\sbsp{n=1}{\infty}{1\over\lambda\sb{n}} = \infty.$ We say that $f\in\Lambda BV$ on an interval (a,b) if $\sum\limits\sbsp{n=1}{\infty}{\vert f(I\sb{n})\vert\over\lambda\sb{n}}<\infty$ for every collection $\{I\sb{n}\}$ of nonoverlapping intervals in (a,b). Here we show that $g\ \circ\ f\in\Lambda BV$ for every $f\in\Lambda BV$ if and only if $g\in Lip1.$ Chaika and Waterman proved an analogous result for the classes GW, UGW and HBV.

In Chapter 3 we prove an analogous theorem for the class $\Phi BV$, which we define as follows: Let $\phi$ be a convex function satisfying $\phi(0) = 0,\ \phi(x)>0$ for $x>0,\ {\phi(x)\over x}\to0$ as $x\to0,$ and ${\phi(x)\over x}\to\infty$ as $x\to\infty.$ We say $f\in\Phi BV$ on (a,b) if $\sum\limits\sbsp{n=1}{\infty}\phi(\vert f(I\sb{n})\vert)<\infty$ for every collection $\{I\sb{n}\}$ of nonoverlapping intervals in (a,b). We will make the further assumption that $\phi$ satisfy the $\Delta\sb2$ condition so that the resulting class $\Phi BV$ forms a linear space. In this chapter we prove that $g\ \circ\ f\in\Phi BV$ for every $f\in\Phi BV$ if and only if $g\in Lip1.$ We also present an interesting condition which is equivalent to the $\Delta\sb2$ condition.

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