## 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.

#### Access

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