Symmetric Properties of Real Functions

Brian thomson

• Publisher: CRC Press
• Year: 2020
• ISBN:
  9781000148336
• Edition: 1
• Topic: Mathematik
• Subtopic: Mathematik Allgemein

This work offers detailed coverage of every important aspect of symmetric structures in function of a single real variable, providing a historical perspective, proofs and useful methods for addressing problems. It provides assistance for real analysis problems involving symmetric derivatives, symmetric continuity and local symmetric structure of sets or functions.

1
The Derivative

1.1 Introduction

The first and second symmetric derivatives of a function f are defined by the expressions

$\begin{array}{c}\text{SD}f\left(x\right)=\underset{t\to 0}{\lim }\frac{f(x+t)-f(x-t)}{2t}\mathrm{.}\\ {\text{SD}}_{2}f\left(x\right)=\underset{t\to 0}{\lim }\frac{f(x+t)+f(x-t)-2f\left(x\right)}{t^2}\mathrm{.}\end{array}$

The first of these is, on occasion, called Lebesgue’s derivative and the second Riemann’s derivative or Schwarz’s derivative. We begin our work with some classical studies of these derivatives. Although our concerns are broader than this, these studies provide us the opportunity to present many of the main themes and will lead us into the subject.

The first question that would obviously arise in a study of the symmetric derivatives is to ask how far properties of ordinary derivatives extend to symmetric derivatives. This is a natural starting point for our investigation and is the starting point taken by Khintchine [151] in 1927. This leads, by relatively familiar methods, to a number of elementary observations. We supplement this with several other important classical, but essentially elementary, results of Schwarz, Mazurkiewicz [197] and [198], and Auerbach [9]. The applications to trigonometric series that appear in Sections 1.4 and 1.7 remain among the central reasons why the subject has attracted attention.

The main result in Khintchine’s fundamental paper, the first really technically interesting result in the subject, appears in Section 1.10. This asks the question: does the existence of the symmetric derivative of a function f on a set E say anything about the ordinary differentiability properties of f on that set? The answer to this question will introduce us to some of the more subtle geometric arguments needed to study effectively this derivative in particular and symmetric properties of real functions in general.

1.2 Even and Odd Properties

1.2.1 The Even and Odd Parts of a Function

A function f is even if f(t) = f(–t) everywhere; a function f is odd if f(t) = –f(–t) everywhere. These conditions represent symmetries in the graph of the function, in the first case symmetry about the y-axis and in the second symmetry about the origin.

At any point x and for any function f we can study these symmetries in the ordinary difference

$\mathit{\Delta }f(x,t)=f(x+t)-f\left(x\right)\mathrm{.}$

This difference can be written as

$f(x+t)-f\left(x\right)=\frac{{\displaystyle f(x+t)-f(x-t)}}{{\displaystyle 2}}+\frac{{\displaystyle f(x+t)+f(x-t)-2f\left(x\right)}}{{\displaystyle 2}}\mathrm{.}$

We shall employ the notations

${{\mathit{\Delta }}^1}_{s}f(x,t)=\mathrm{f...}$