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
• Category: Library

✦ Synopsis

This monographis a self-contained exposition of the definition and properties of functions of bounded variation and their various generalizations; the analytical properties of nonlinear composition operators in spaces of such functions; applications to Fourier analysis, nonlinear integral equations, and boundary value problems. The book is written for non-specialists. Every chapter closes with a list of exercises and open problems

✦ Table of Contents

Content: 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 variations2.7 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 44.6 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 series7.2 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.
Abstract: This monographis a self-contained exposition of the definition and properties of functionsof bounded variation and their various generalizations; the analytical properties of nonlinear composition operators in spaces of such functions; applications to Fourier analysis, nonlinear integral equations, and boundary value problems. The book is written for non-specialists. Every chapter closes with a list of exercises and open problems