Bounded Variation and Around

by Jürgen Appell, Józef Banas, Nelson José Merentes Díaz

Overview

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.

Product Details

ISBN-13:	9783110265071
Publisher:	De Gruyter
Publication date:	12/12/2013
Series:	De Gruyter Series in Nonlinear Analysis and Applications , #17
Pages:	486
Product dimensions:	6.69(w) x 9.45(h) x 0.05(d)
Age Range:	18 Years

About the Author

Jürgen Appell, University of Würzburg, Germany; Jozef Banas, Technical University of Rzeszow, Poland; Nelson José Merentes Díaz, Central University of Venezuela, Caracas, Venezuela.

Table of Contents

Preface v

Introduction 1

0 Prerequisites 7

0.1 The Lebesgue integral 7

0.2 Some functional analysis 18

0.3 Basic function spaces 25

0.4 Comments on Chapter 0 43

0.5 Exercises to Chapter 0 46

1 Classical BV-spaces 55

1.1 Functions of bounded variation 55

1.2 Bounded variation and continuity 71

1.3 Functions of bounded Wiener variation 84

1.4 Functions of several variables 91

1.5 Comments on Chapter 1 100

1.6 Exercises to Chapter 1 104

2 Nonclassical BV-spaces 112

2.1 The Wiener-Youngvariation 112

2.2 The Waterman variation 125

2.3 The Schramm variation 152

2.4 The Riesz-Medvedev variation 161

2.5 The Korenblum variation 169

2.6 Higher order Wiener-type variations 182

2.7 Comments on Chapter 2 187

2.8 Exercises to Chapter 2 202

3 Absolutely continuous functions 208

3.1 Continuity and absolute continuity 208

3.2 The Vitali-Banach-Zaretskij theorem 211

3.3 Reconstructing a function from its derivative 218

3.4 Rectifiable functions 231

3.5 The Riesz-Medvedev theorem 240

3.6 Higher order Riesz-type variations 244

3.7 Comments on Chapter 3 249

3.8 Exercises to Chapter 3 260

4 Riemann-Stieltjes integrals 268

4.1 Classical RS-integrals 268

4.2 Bounded variation and duality 292

4.3 Bounded p-variation and duality 298

4.4 Nonclassical RS-integrals 302

4.5 Comments on Chapter 4 311

4.6 Exercises to Chapter 4 316

5 Nonlinear composition operators 324

5.1 The composition operator problem 324

5.2 Boundedness and continuity 344

5.3 Spaces of differentiable functions 354

5.4 Global Lipschitz continuity 364

5.5 Local Lipschitz continuity 368

5.6 Comments on Chapter 5 377

5.7 Exercises to Chapter 5 382

6 Nonlinear superposition operators 385

6.1 Boundedness and continuity 385

6.2 Lipschitz continuity 400

6.3 Uniform boundedness and continuity 406

6.4 Functions of several variables 415

6.5 Comments on Chapter 6 419

6.6 Exercises to Chapter 6 423

7 Some applications 425

7.1 Convergence criteria for Fourier series 425

7.2 Fourier series and Waterman spaces 429

7.3 Applications to nonlinear integral equations 435

7.4 Comments on Chapter7 444

References 453

List of functions 467

List of symbols 468

Index 472

Editorial Reviews

"This very interesting and well-written monograph will be useful for students and scientists." Mathematical Reviews

"The book is self-contained and presents a deep description of this class of functions. It can be very useful for beginners in analysis, as well as for experts in certain specific areas. […] The book is very concrete and is a very good combination of deep rigorous results and and detailed explanation. Partly, the book can be included into the educational process and surely will be a table-book for many analysts." Zentralblatt für Mathematik

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