Functional Analysis is based on the lecture notes of distinguished authors and is designed to cater to the needs of students who are yet to be exposed to the subject, as well as senior undergraduate- and graduate-level students at universities the world over. The text begins with a preliminary chapter that establishes uniform notations and covers background material in real analysis, linear algebra, and metric spaces. It is followed by chapters on Normed and Banach Spaces, Bounded Linear Operators and Bounded Linear Functional. This text also deals with the concept and specific geometry of Hilbert Spaces, Functional and Operators on Hilbert Spaces, and an Introduction to Spectral Theory. The appendix provides an introduction to Schauder Bases. This is a second edition, written in a more simple and lucid language and illustrated with familiar examples. It is an ideal textbook for easy comprehension of the subject. The clear explanations, numerous examples, problems and illustrative figures also make the text invaluable for self-study and as a reference book.

Extremum Seeking through Delays and PDEs, the first book on the topic, expands the scope of applicability of the extremum seeking method, from static and finite-dimensional systems to infinite-dimensional systems. Readers will find: Numerous algorithms for model-free real-time optimization are developed and their convergence guaranteed. Extensions from single-player optimization to noncooperative games, under delays and pdes, are provided. The delays and pdes are compensated in the control designs using the pde backstepping approach, and stability is ensured using infinite-dimensional versions of averaging theory. Accessible and powerful tools for analysis. This book is intended for control engineers in all disciplines (electrical, mechanical, aerospace, chemical), mathematicians, physicists, biologists, and economists. It is appropriate for graduate students, researchers, and industrial users.

Double Sequence Spaces and Four-Dimensional Matrices provides readers with a clear introduction to the spaces of double sequences and series, as well as their properties. The book then goes beyond this to investigate paranormed double sequence spaces and their algebraic and topological properties, triangle matrices and their domains in certain spaces of double sequences, dual spaces of double sequence spaces, and matrix transformations between double sequence spaces and related topics. Each chapter contains a conclusion section highlighting the importance of results and pointing out possible new ideas that can be studied further. Features Suitable for students at graduate or post-graduate level and researchers Investigates different types of summable spaces and computes their duals Characterizes several four-dimensional matrix classes transforming one summable space into other Discusses several algebraic and topological properties of new sequence spaces generated by the domain of triangles.

1) Includes exemplary MATLAB codes 2) Provides a comprehensive foundation in Fourier methods, essential for a mathematical approach to engineering 3) Applies MFS to hot topics in the field: multi-domain, multi- physics, and multi-scale characteristics 4) Applies Fourier method to structural vibrations, acoustics and vibro-acoustic 5) Aids engineers in solving boundary value problems and differential equations

This book helps students explore Fourier analysis and its related topics, helping them appreciate why it pervades many fields of mathematics, science, and engineering. This introductory textbook was written with mathematics, science, and engineering students with a background in calculus and basic linear algebra in mind. It can be used as a textbook for undergraduate courses in Fourier analysis or applied mathematics, which cover Fourier series, orthogonal functions, Fourier and Laplace transforms, and an introduction to complex variables. These topics are tied together by the application of the spectral analysis of analog and discrete signals, and provide an introduction to the discrete Fourier transform. A number of examples and exercises are provided including implementations of Maple, MATLAB, and Python for computing series expansions and transforms. After reading this book, students will be familiar with: * Convergence and summation of infinite series * Representation of functions by infinite series * Trigonometric and Generalized Fourier series * Legendre, Bessel, gamma, and delta functions * Complex numbers and functions * Analytic functions and integration in the complex plane * Fourier and Laplace transforms. * The relationship between analog and digital signals Dr. Russell L. Herman is a professor of Mathematics and Professor of Physics at the University of North Carolina Wilmington. A recipient of several teaching awards, he has taught introductory through graduate courses in several areas including applied mathematics, partial differential equations, mathematical physics, quantum theory, optics, cosmology, and general relativity. His research interests include topics in nonlinear wave equations, soliton perturbation theory, fluid dynamics, relativity, chaos and dynamical systems.

Principles of Analysis: Measure, Integration, Functional Analysis, and Applications prepares readers for advanced courses in analysis, probability, harmonic analysis, and applied mathematics at the doctoral level. The book also helps them prepare for qualifying exams in real analysis. It is designed so that the reader or instructor may select topics suitable to their needs. The author presents the text in a clear and straightforward manner for the readers' benefit. At the same time, the text is a thorough and rigorous examination of the essentials of measure, integration and functional analysis. The book includes a wide variety of detailed topics and serves as a valuable reference and as an efficient and streamlined examination of advanced real analysis. The text is divided into four distinct sections: Part I develops the general theory of Lebesgue integration; Part II is organized as a course in functional analysis; Part III discusses various advanced topics, building on material covered in the previous parts; Part IV includes two appendices with proofs of the change of the variable theorem and a joint continuity theorem. Additionally, the theory of metric spaces and of general topological spaces are covered in detail in a preliminary chapter . Features: Contains direct and concise proofs with attention to detail Features a substantial variety of interesting and nontrivial examples Includes nearly 700 exercises ranging from routine to challenging with hints for the more difficult exercises Provides an eclectic set of special topics and applications About the Author: Hugo D. Junghenn is a professor of mathematics at The George Washington University. He has published numerous journal articles and is the author of several books, including Option Valuation: A First Course in Financial Mathematics and A Course in Real Analysis. His research interests include functional analysis, semigroups, and probability.

Special functions play a very important role in solving various families of ordinary and partial differential equations as well as their fractional-order analogs which model real-life situations. Due to the non-local nature and memory effect, fractional calculus is capable of modeling many situations which arise in engineering. This book includes a collection of related topics associated with such equations and their relevance and significance in engineering. Special Functions in Fractional Calculus and Engineering highlights the significance and applicability of special functions in solving fractional-order differential equations with engineering applications. The book focuses on the non-local nature and memory effect of fractional calculus in modeling relevant to engineering science and covers a variety of important and useful methods using special functions for solving various types of fractional-order models relevant to engineering science. The book goes on to illustrate the applicability and usefulness of special functions by justifying their numerous and widespread occurrences in the solution of fractional-order differential, integral, and integrodifferential equations. The book holds a wide variety of interconnected fundamental and advanced topics with interdisciplinary applications that combine applied mathematics and engineering sciences. useful to graduate students, Ph.D. scholars, researchers, and educators interested in special functions, fractional calculus, mathematical modeling, and engineering. .

Fundamental to the study of any mathematical structure is an understanding of its symmetries. In the class of Banach spaces, this leads naturally to a study of isometries-the linear transformations that preserve distances. In his foundational treatise, Banach showed that every linear isometry on the space of continuous functions on a compact metric space must transform a continuous function x into a continuous function y satisfying y(t) = h(t)x(p(t)), where p is a homeomorphism and |h| is identically one.

Isometries on Banach Spaces: Function Spaces is the first of two planned volumes that survey investigations of Banach-space isometries. This volume emphasizes the characterization of isometries and focuses on establishing the type of explicit, canonical form given above in a variety of settings. After an introductory discussion of isometries in general, four chapters are devoted to describing the isometries on classical function spaces. The final chapter explores isometries on Banach algebras.

This treatment provides a clear account of historically important results, exposes the principal methods of attack, and includes some results that are more recent and some that are lesser known. Unique in its focus, this book will prove useful for experts as well as beginners in the field and for those who simply want to acquaint themselves with this area of Banach space theory.

Wavelets from a Statistical Perspective offers a modern, 2nd generation look on wavelets, far beyond the rigid setting of the equispaced, dyadic wavelets in the early days. With the methods of this book, based on the lifting scheme, researchers can set up a wavelet or another multiresolution analysis adapted to their data, ranging from images to scattered data or other irregularly spaced observations. Whereas classical wavelets stand a bit apart from other nonparametric methods, this book adds a multiscale touch to your spline, kernel or local polynomial smoothing procedure, thereby extending its applicability to nonlinear, nonparametric processing for piecewise smooth data. One of the chapters of the book constructs B-spline wavelets on nonequispaced knots and multiscale local polynomial transforms. In another chapter, the link between wavelets and Fourier analysis, ubiquitous in the classical approach, is explained, but without being inevitable. In further chapters the discrete wavelet transform is contrasted with the continuous version, the nondecimated (or maximal overlap) transform taking an intermediate position. An important principle in designing a wavelet analysis through the lifting scheme is finding the right balance between bias and variance. Bias and variance also play a crucial role in the nonparametric smoothing in a wavelet framework, in finding well working thresholds or other smoothing parameters. The numerous illustrations can be reproduced with the online available, accompanying software. The software and the exercises can also be used as a starting point in the further exploration of the material.

This Research Note addresses several pivotal problems in spectral theory and nonlinear functional analysis in connection with the analysis of the structure of the set of zeroes of a general class of nonlinear operators. It features the construction of an optimal algebraic/analytic invariant for calculating the Leray-Schauder degree, new methods for solving nonlinear equations in Banach spaces, and general properties of components of solutions sets presented with minimal use of topological tools. The author also gives several applications of the abstract theory to reaction diffusion equations and systems.

The results presented cover a thirty-year period and include recent, unpublished findings of the author and his coworkers. Appealing to a broad audience, Spectral Theory and Nonlinear Functional Analysis contains many important contributions to linear algebra, linear and nonlinear functional analysis, and topology and opens the door for further advances.

Multiplicative Differential Equations: Volume 2 is the second part of a comprehensive approach to the subject. It continues a series of books written by the authors on multiplicative, geometric approaches to key mathematical topics. This volume is devoted to the theory of multiplicative differential systems. The asymptotic behavior of the solutions of such systems is studied. Stability theory for multiplicative linear and nonlinear systems is introduced and boundary value problems for second order multiplicative linear and nonlinear equations are explored. The authors also present first order multiplicative partial differential equations. Each chapter ends with a section of practical problems. The book is accessible to graduate students and researchers in mathematics, physics, engineering and biology.

For those who have a background in advanced calculus, elementary topology and functional analysis - from applied mathematicians and engineers to physicists - researchers and graduate students alike - this work provides a comprehensive analysis of the many important integral transforms and renders particular attention to all of the technical aspects of the subject. The author presents the last two decades of research and includes important results from other works.
With a description in Chapter 1 of foundational, topological information for the study of the distributions (generalized functions) and ultradistributions to be considered, including the structure of these distributions, the author then moves on to an analysis of the transforms and ends with a discussion of the eigenfunction expansion of generalized functions. Applications of the transforms to various mathematical, physical, and engineering problems are given, and open research problems present a challenge to the reader. An extensi

This new edition of an indispensable text provides a clear treatment of Fourier Series, Fourier Transforms, and FFTs. The unique software, included with the book and newly updated for this edition, allows the reader to generate, firsthand, images of all aspects of Fourier analysis described in the text. Topics covered include applications to vibrating strings, heat conduction, removal of noise and frequency detection, filtering of Fourier Series and improvement of convergence, and much more.

This volume is essentially a self-contained presentation of the theory of reproducing kernels in connection with integral transforms in the framework of Hilbert spaces. It is a general and fundamental concept and a potentially powerful theory combined with the integral transforms. A variety of concrete results of its application are given systematically for isometrical identities and inversion formulas for various typical integral transforms, best approximation theories of functions, analytic extension formulas, real inversion formulas for the Laplace transform, inverse source problems, representationbs of inverse functions, natural norm inequalities in nonlinear transforms and stability of Lipschitz type in determintion of initial heat distribution.

With the aim to better understand nature, mathematical tools are being used nowadays in many different fields. The concept of integral transforms, in particular, has been found to be a useful mathematical tool for solving a variety of problems not only in mathematics, but also in various other branches of science, engineering, and technology. Integral Transforms and Engineering: Theory, Methods, and Applications presents a mathematical analysis of integral transforms and their applications. The book illustrates the possibility of obtaining transfer functions using different integral transforms, especially when mapping any function into the frequency domain. Various differential operators, models, and applications are included such as classical derivative, Caputo derivative, Caputo-Fabrizio derivative, and Atangana-Baleanu derivative. This book is a useful reference for practitioners, engineers, researchers, and graduate students in mathematics, applied sciences, engineering, and technology fields.

Combining both classical and current methods of analysis, this text present discussions on the application of functional analytic methods in partial differential equations. It furnishes a simplified, self-contained proof of Agmon-Douglis-Niremberg's Lp-estimates for boundary value problems, using the theory of singular integrals and the Hilbert transform.

Understanding Real Analysis, Second Edition offers substantial coverage of foundational material and expands on the ideas of elementary calculus to develop a better understanding of crucial mathematical ideas. The text meets students at their current level and helps them develop a foundation in real analysis. The author brings definitions, proofs, examples and other mathematical tools together to show how they work to create unified theory. These helps students grasp the linguistic conventions of mathematics early in the text. The text allows the instructor to pace the course for students of different mathematical backgrounds. Key Features: Meets and aligns with various student backgrounds Pays explicit attention to basic formalities and technical language Contains varied problems and exercises Drives the narrative through questions

Typically, undergraduates see real analysis as one of the most difficult courses that a mathematics major is required to take. The main reason for this perception is twofold: Students must comprehend new abstract concepts and learn to deal with these concepts on a level of rigor and proof not previously encountered. A key challenge for an instructor of real analysis is to find a way to bridge the gap between a student's preparation and the mathematical skills that are required to be successful in such a course. Real Analysis: With Proof Strategies provides a resolution to the "bridging-the-gap problem." The book not only presents the fundamental theorems of real analysis, but also shows the reader how to compose and produce the proofs of these theorems. The detail, rigor, and proof strategies offered in this textbook will be appreciated by all readers. Features Explicitly shows the reader how to produce and compose the proofs of the basic theorems in real analysis Suitable for junior or senior undergraduates majoring in mathematics.

The equations of mathematical physics are the mathematical models of the large class of phenomenon of physics, chemistry, biology, economics, etc. In Sequential Models of Mathematical Physics, the author considers the justification of the process of constructing mathematical models. The book seeks to determine the classic, generalized and sequential solutions, the relationship between these solutions, its direct physical sense, the methods of its practical finding, and its existence. Features Describes a sequential method based on the construction of space completion, as well as its applications in number theory, the theory of distributions, the theory of extremum, and mathematical physics Presentation of the material is carried out on the simplest example of a one-dimensional stationary heat transfer process; all necessary concepts and constructions are introduced and illustrated with elementary examples, which makes the material accessible to a wide area of readers The solution of a specific mathematical problem is obtained as a result of the joint application of methods and concepts from completely different mathematical directions

This monograph has two main purposes, first to act as a companion volume to more advanced texts by gathering together the principal mathematical topics commonly used in developing scattering theories and, in so doing, provide a reasonable, self-contained introduction to linear and nonlinear scattering theory for those who might wish to begin working in the area. Secondly, to indicate how these various aspects might be applied to problems in mathematical physics and the applied sciences. Of particular interest will be the influence of boundary conditions.

The book features original chapters on sequence spaces involving the idea of ideal convergence, modulus function, multiplier sequences, Riesz mean, Fibonacci difference matrix etc., and illustrate their involvement in various applications. The preliminaries have been presented in the beginning of each chapter and then the advanced discussion takes place, so it is useful for both expert and nonexpert on aforesaid topics. The book consists of original thirteen research chapters contributed by the well-recognized researchers in the field of sequence spaces with associated applications. Features Discusses the Fibonacci and vector valued difference sequence spaces Presents the solution of Volterra integral equation in Banach algebra Discusses some sequence spaces involving invariant mean and related to the domain of Jordan totient matrix Presents the Tauberian theorems of double sequences Discusses the paranormed Riesz difference sequence space of fractional order Includes a technique for studying the existence of solutions of infinite system of functional integro-differential equations in Banach sequence spaces The subject of book is an active area of research of present time internationally and would serve as a good source for researcher and educators involved with the topic of sequence spaces.

Features Suitable for graduate students and professional researchers in operator theory and/or analysis Numerous applications in related scientific fields and areas.

This book provides an introduction to functional analysis for non-experts in mathematics. As such, it is distinct from most other books on the subject that are intended for mathematicians. Concepts are explained concisely with visual materials, making it accessible for those unfamiliar with graduate-level mathematics. Topics include topology, vector spaces, tensor spaces, Lebesgue integrals, and operators, to name a few. Each chapter explains, concisely, the purpose of the specific topic and the benefit of understanding it. Researchers and graduate students in physics, mechanical engineering, and information science will benefit from this view of functional analysis.

The book discusses basic concepts of functional analysis, measure and integration theory, calculus of variations and duality and its applications to variational problems of non-convex nature, such as the Ginzburg-Landau system in superconductivity, shape optimization models, dual variational formulations for micro-magnetism and others. Numerical Methods for such and similar problems, such as models in flight mechanics and the Navier-Stokes system in fluid mechanics have been developed through the generalized method of lines, including their matrix finite dimensional approximations. It concludes with a review of recent research on Riemannian geometry applied to Quantum Mechanics and Relativity. The book will be of interest to applied mathematicians and graduate students in applied mathematics. Physicists, engineers and researchers in related fields will also find the book useful in providing a mathematical background applicable to their respective professional areas.

Fixed Point Results in W-Distance Spaces is a self-contained and comprehensive reference for advanced fixed-point theory and can serve as a useful guide for related research. The book can be used as a teaching resource for advanced courses on fixed-point theory, which is a modern and important field in mathematics. It would be especially valuable for graduate and postgraduate courses and seminars. Features Written in a concise and fluent style, covers a broad range of topics and includes related topics from research. Suitable for researchers and postgraduates. Contains brand new results not published elsewhere.