Industrial and Applied Mathematics Abul Hasan Siddiqi Functional Analysis and Applications Industrial and Applied Mathematics Editor-in-chief Abul Hasan Siddiqi, Sharda University, Greater Noida, India Editorial Board Zafer Aslan, International Centre for Theoretical Physics, Istanbul, Turkey M. Brokate, Technical University, Munich, Germany N.K. Gupta, Indian Institute of Technology Delhi, New Delhi, India Akhtar Khan, Center for Applied and Computational Mathematics, Rochester, USA Rene Lozi, University of Nice Sophia-Antipolis, Nice, France Pammy Manchanda, Guru Nanak Dev University, Amritsar, India M. Zuhair Nashed, University of Central Florida, Orlando, USA Govindan Rangarajan, Indian Institute of Science, Bengaluru, India K.R. Sreenivasan, Polytechnic School of Engineering, New York, USA The Industrial and Applied Mathematics series publishes high-quality research-level monographs, lecture notes and contributed volumes focusing on areas where mathematics is used in a fundamental way, such as industrial mathematics, bio-mathematics, financial mathematics, applied statistics, operations research and computer science. More information about this series at http://www.springer.com/series/13577 Abul Hasan Siddiqi Functional Analysis and Applications 123 Abul Hasan Siddiqi School of Basic Sciences and Research Sharda University Greater Noida, Uttar Pradesh India ISSN 2364-6837 ISSN 2364-6845 (electronic) Industrial and Applied Mathematics ISBN 978-981-10-3724-5 ISBN 978-981-10-3725-2 (eBook) https://doi.org/10.1007/978-981-10-3725-2 Library of Congress Control Number: 2018935211 © Springer Nature Singapore Pte Ltd. 2018 This work is subject to copyright. 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Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. part of Springer Nature The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore To My wife Azra Preface Functional analysis was invented and developed in the twentieth century. Besides being an area of independent mathematical interest, it provides many fundamental notions essential for modeling, analysis, numerical approximation, and computer simulation processes of real-world problems. As science and technology are increasingly refined and interconnected, the demand for advanced mathematics beyond the basic vector algebra and differential and integral calculus has greatly increased. There is no dispute on the relevance of functional analysis; however, there have been differences of opinion among experts about the level and methodology of teaching functional analysis. In the recent past, its applied nature has been gaining ground. The main objective of this book is to present all those results of functional analysis, which have been frequently applied in emerging areas of science and technology. Functional analysis provides basic tools and foundation for areas of vital importance such as optimization, boundary value problems, modeling real-world phenomena, finite and boundary element methods, variational equations and inequalities, inverse problems, and wavelet and Gabor analysis. Wavelets, formally invented in the mid-eighties, have found significant applications in image pro- cessing and partial differential equations. Gabor analysis was introduced in 1946, gaining popularity since the last decade among the signal processing community and mathematicians. The book comprises 15 chapters, an appendix, and a comprehensive updated bibliography. Chapter 1 is devoted to basic results of metric spaces, especially an important fixed-point theorem called the Banach contraction mapping theorem, and its applications to matrix, integral, and differential equations. Chapter 2 deals with basic definitions and examples related to Banach spaces and operators defined on such spaces. A sufficient number of examples are presented to make the ideas clear. Algebras of operators and properties of convex functionals are discussed. Hilbert space, an infinite-dimensional analogue of Euclidean space of finite dimension, is introduced and discussed in detail in Chap. 3. In addition, important results such as projection theorem, Riesz representation theorem, properties of self-adjoint, vii viii Preface positive, normal, and unitary operators, relationship between bounded linear operator and bounded bilinear form, and Lax–Milgram lemma dealing with the existence of solutions of abstract variational problems are presented. Applications and generalizations of the Lax–Milgram lemma are discussed in Chaps. 7 and 8. Chapter 4 is devoted to the Hahn–Banach theorem, Banach–Alaoglu theorem, uniform boundedness principle, open mapping, and closed graph theorems along with the concept of weak convergence and weak topologies. Chapter 5 provides an extension of finite-dimensional classical calculus to infinite-dimensional spaces, which is essential to understand and interpret various current developments of science and technology. More precisely, derivatives in the sense of Gâteau, Fréchet, Clarke (subgradient), and Schwartz (distributional derivative) along with Sobolev spaces are the main themes of this chapter. Fundamental results concerning exis- tence and uniqueness of solutions and algorithm for finding solutions of opti- mization problems are described in Chap. 6. Variational formulation and existence of solutions of boundary value problems representing physical phenomena are described in Chap. 7. Galerkin and Ritz approximation methods are also included. Finite element and boundary element methods are introduced and several theorems concerning error estimation and convergence are proved in Chap. 8. Chapter 9 is devoted to variational inequalities. A comprehensive account of this elegant mathematical model in terms of operators is given. Apart from existence and uniqueness of solutions, error estimation and finite element methods for approxi- mate solutions and parallel algorithms are discussed. The chapter is mainly based on the work of one of its inventors, J. L. Lions, and his co-workers and research students. Activities at the Stampacchia School of Mathematics, Erice, Italy, are providing impetus to researchers in this field. Chapter 10 is devoted to rudiments of spectral theory with applications to inverse problems. We present frame and basis theory in Hilbert spaces in Chap. 11. Chapter 12 deals with wavelets. Broadly, wavelet analysis is a refinement of Fourier analysis and has attracted the attention of researchers in mathematics, physics, and engineering alike. Replacement of the classical Fourier methods, wherever they have been applied, by emerging wavelet methods has resulted in drastic improvements. In this chapter, a detailed account of this exciting theory is presented. Chapter 13 presents an introduction to applications of wavelet methods to partial differential equations and image processing. These are emerging areas of current interest. There is still a wide scope for further research. Models and algorithms for removal of an unwanted component (noise) of a signal are discussed in detail. Error estimation of a given image with its wavelet repre- sentation in the Besov norm is given. Wavelet frames are comparatively a new addition to wavelet theory. We discuss their basic properties in Chap. 14. Dennis Gabor, Nobel Laureate of Physics (1971), introduced windowed Fourier analysis, now called Gabor analysis, in 1946. Fundamental concepts of this analysis with certain applications are presented in Chap. 15. In appendix, we present a resume of the results of topology, real analysis, calculus, and Fourier analysis which we often use in this book. Chapters 9, 12, 13, and 15 contain recent results opening up avenues for further work. Preface ix The book is self-contained and provides examples, updated references, and applications in diverse fields. Several problems are thought-provoking, and many lead to new results and applications. The book is intended to be a textbook for graduate or senior undergraduate students in mathematics. It could also be used for an advance course in system engineering, electrical engineering, computer engi- neering, and management sciences. The proofs of theorems and other items marked with an asterisk may be omitted for a senior undergraduate course or a course in other disciplines. Those who are mainly interested in applications of wavelets and