Bounded Variation and Around

✍ Scribed by Jürgen Appell; Józef Banas; Nelson José Merentes Díaz

• Publisher: De Gruyter
• Year: 2013
• Tongue: English
• Leaves: 488
• Series: De Gruyter Series in Nonlinear Analysis and Applications; 17
• Category: Library

✦ Synopsis

The aim of this monograph is to give a thorough and self-contained account of functions of (generalized) bounded variation, the methods connected with their study, their relations to other important function classes, and their applications to various problems arising in Fourier analysis and nonlinear analysis.

In the first part the basic facts about spaces of functions of bounded variation and related spaces are collected, the main ideas which are useful in studying their properties are presented, and a comparison of their importance and suitability for applications is provided, with a particular emphasis on illustrative examples and counterexamples. The second part is concerned with (sometimes quite surprising) properties of nonlinear composition and superposition operators in such spaces. Moreover, relations with Riemann-Stieltjes integrals, convergence tests for Fourier series, and applications to nonlinear integral equations are discussed.

The only prerequisite for understanding this book is a modest background in real analysis, functional analysis, and operator theory. It is addressed to non-specialists who want to get an idea of the development of the theory and its applications in the last decades, as well as a glimpse of the diversity of the directions in which current research is moving. Since the authors try to take into account recent results and state several open problems, this book might also be a fruitful source of inspiration for further research.

✦ Table of Contents

Preface
Introduction
0 Prerequisites
0.1 The Lebesgue integral
0.2 Some functional analysis
0.3 Basic function spaces
0.4 Comments on Chapter 0
0.5 Exercises to Chapter 0
1 Classical BV-spaces
1.1 Functions of bounded variation
1.2 Bounded variation and continuity
1.3 Functions of bounded Wiener variation
1.4 Functions of several variables
1.5 Comments on Chapter 1
1.6 Exercises to Chapter 1
2 Nonclassical BV-spaces
2.1 The Wiener–Young variation
2.2 The Waterman variation
2.3 The Schramm variation
2.4 The Riesz–Medvedev variation
2.5 The Korenblum variation
2.6 Higher order Wiener-type variations
2.7 Comments on Chapter 2
2.8 Exercises to Chapter 2
3 Absolutely continuous functions
3.1 Continuity and absolute continuity
3.2 The Vitali–Banach–Zaretskij theorem
3.3 Reconstructing a function from its derivative
3.4 Rectifiable functions
3.5 The Riesz–Medvedev theorem
3.6 Higher order Riesz-type variations
3.7 Comments on Chapter 3
3.8 Exercises to Chapter 3
4 Riemann–Stieltjes integrals
4.1 Classical RS-integrals
4.2 Bounded variation and duality
4.3 Bounded p-variation and duality
4.4 Nonclassical RS-integrals
4.5 Comments on Chapter 4
4.6 Exercises to Chapter 4
5 Nonlinear composition operators
5.1 The composition operator problem
5.2 Boundedness and continuity
5.3 Spaces of differentiable functions
5.4 Global Lipschitz continuity
5.5 Local Lipschitz continuity
5.6 Comments on Chapter 5
5.7 Exercises to Chapter 5
6 Nonlinear superposition operators
6.1 Boundedness and continuity
6.2 Lipschitz continuity
6.3 Uniform boundedness and continuity
6.4 Functions of several variables
6.5 Comments on Chapter 6
6.6 Exercises to Chapter 6
7 Some applications
7.1 Convergence criteria for Fourier series
7.2 Fourier series and Waterman spaces
7.3 Applications to nonlinear integral equations
7.4 Comments on Chapter 7
References
List of functions
List of symbols
Index