Funktionsanalyse og transformation

This textbook is an introduction to the methods needed to solve partial differential equations (PDEs). Readers are introduced to PDEs that come from a variety of fields in engineering and the natural sciences. The chapters include the following topics: First Order PDEs, Second Order PDEs, Fourier Series, Separation of Variables, the Fourier Transform, and higher dimensional problems. Readers are guided through these chapters where techniques for solving first and second order PDEs are introduced. Each chapter ends with series of exercises to facilitate learning as well as illustrate the material presented in each chapter.

The purpose of this monograph is to provide a systematic account of the theory of noncommutative integration in semi-finite von Neumann algebras. It is designed to serve as an introductory graduate level text as well as a basic reference for more established mathematicians with interests in the continually expanding areas of noncommutative analysis and probability. Its origins lie in two apparently distinct areas of mathematical analysis: the theory of operator ideals going back to von Neumann and Schatten and the general theory of rearrangement invariant Banach lattices of measurable functions which has its roots in many areas of classical analysis related to the well-known Lp-spaces. A principal aim, therefore, is to present a general theory which contains each of these motivating areas as special cases.

This book compiles research and surveys devoted to the areas of mathematical analysis, approximation theory, and optimization. Being dedicated to A.-M. Legendre's work, contributions to this volume are devoted to those branches of mathematics and its applications that have been influenced, directly or indirectly, by the mathematician. Additional contributions provide a historical background as it relates to Legendre's work and its association to the foundation of Greece's higher education.Topics covered in this book include the investigation of the Jensen-Steffensen inequality, Ostrowski and trapezoid type inequalities, a Hilbert-Type Inequality, Hardy¿s inequality, dynamic unilateral contact problems, square-free values of a category of integers, a maximum principle for general nonlinear operators, the application of Ergodic Theory to an alternating series expansion for real numbers, bounds for similarity condition numbers of unbounded operators, finite element methods with higher order polynomials, generating functions for the Fubini type polynomials, local asymptotics for orthonormal polynomials, trends in geometric function theory, quasi variational inclusions, Kleene fixed point theorems, ergodic states, spontaneous symmetry breaking and quasi-averages.It is hoped that this book will be of interest to a wide spectrum of readers from several areas of pure and applied sciences, and will be useful to undergraduate students, graduate level students, and researchers who want to be kept up to date on the results and theories in the subjects covered in this volume.

This textbook introduces the study of partial differential equations using both analytical and numerical methods. By intertwining the two complementary approaches, the authors create an ideal foundation for further study. Motivating examples from the physical sciences, engineering, and economics complete this integrated approach.A showcase of models begins the book, demonstrating how PDEs arise in practical problems that involve heat, vibration, fluid flow, and financial markets. Several important characterizing properties are used to classify mathematical similarities, then elementary methods are used to solve examples of hyperbolic, elliptic, and parabolic equations. From here, an accessible introduction to Hilbert spaces and the spectral theorem lay the foundation for advanced methods. Sobolev spaces are presented first in dimension one, before being extended to arbitrary dimension for the study of elliptic equations. An extensive chapter on numerical methods focuses onfinite difference and finite element methods. Computer-aided calculation with Maple¿ completes the book. Throughout, three fundamental examples are studied with different tools: Poisson¿s equation, the heat equation, and the wave equation on Euclidean domains. The Black¿Scholes equation from mathematical finance is one of several opportunities for extension.Partial Differential Equations offers an innovative introduction for students new to the area. Analytical and numerical tools combine with modeling to form a versatile toolbox for further study in pure or applied mathematics. Illuminating illustrations and engaging exercises accompany the text throughout. Courses in real analysis and linear algebra at the upper-undergraduate level are assumed.

Large numbers of studies of the KdV equation have appeared since the pioneering paper by Gardner, Greene, Kruskal, and Miura in 1967. Most of those works have employed the inverse spectral method for 1D Schrödinger operators or an advanced Fourier analysis. Although algebraic approaches have been discovered by HirotäSato and Marchenko independently, those have not been fully investigated and analyzed. The present book offers a new approach to the study of the KdV equation, which treats decaying initial data and oscillating data in a unified manner. The author¿s method is to represent the tau functions introduced by HirotäSato and developed by Segal¿Wilson later in terms of the Weyl¿Titchmarsh functions (WT functions, in short) for the underlying Schrödinger operators. The main result is stated by a class of WT functions satisfying some of the asymptotic behavior along a curve approaching the spectrum of the Schrödinger operators at +¿ in an order of -(n-1/2)for the nth KdV equation. This class contains many oscillating potentials (initial data) as well as decaying ones. Especially bounded smooth ergodic potentials are included, and under certain conditions on the potentials, the associated Schrödinger operators have dense point spectrum. This provides a mathematical foundation for the study of the soliton turbulence problem initiated by Zakharov, which was the author¿s motivation for extending the class of initial data in this book. A large class of almost periodic potentials is also included in these ergodic potentials. P. Deift has conjectured that any solutions to the KdV equation starting from nearly periodic initial data are almost periodic in time. Therefore, our result yields a foundation for this conjecture. For the reader¿s benefit, the author has included here (1) a basic knowledge of direct and inverse spectral problem for 1D Schrödinger operators, including the notion of the WT functions; (2)Satös Grassmann manifold method revised by Segal¿Wilson; and (3) basic results of ergodic Schrödinger operators.

This text covers a first course in bilinear maps and tensor products intending to bring the reader from the beginning of functional analysis to the frontiers of exploration with tensor products. Tensor products, particularly in infinite-dimensional normed spaces, are heavily based on bilinear maps. The author brings these topics together by using bilinear maps as an auxiliary, yet fundamental, tool for accomplishing a consistent, useful, and straightforward theory of tensor products. The author¿s usual clear, friendly, and meticulously prepared exposition presents the material in ways that are designed to make grasping concepts easier and simpler. The approach to the subject is uniquely presented from an operator theoretic view. An introductory course in functional analysis is assumed. In order to keep the prerequisites as modest as possible, there are two introductory chapters, one on linear spaces (Chapter 1) and another on normed spaces (Chapter 5), summarizing the background material required for a thorough understanding. The reader who has worked through this text will be well prepared to approach more advanced texts and additional literature on the subject.The book brings the theory of tensor products on Banach spaces to the edges of Grothendieck's theory, and changes the target towards tensor products of bounded linear operators. Both Hilbert-space and Banach-space operator theory are considered and compared from the point of view of tensor products. This is done from the first principles of functional analysis up to current research topics, with complete and detailed proofs. The first four chapters deal with the algebraic theory of linear spaces, providing various representations of the algebraic tensor product defined in an axiomatic way. Chapters 5 and 6 give the necessary background concerning normed spaces and bounded bilinear mappings. Chapter 7 is devoted to the study of reasonable crossnorms on tensor product spaces, discussing in detail the important extreme realizations of injective and projective tensor products. In Chapter 8 uniform crossnorms are introduced in which the tensor products of operators are bounded; special attention is paid to the finitely generated situation. The concluding Chapter 9 is devoted to the study of the Hilbert space setting and the spectral properties of the tensor products of operators. Each chapter ends with a section containing ¿Additional Propositions" and suggested readings for further studies.

Over the course of his distinguished career, Jörg Eschmeier made a number of fundamental contributions to the development of operator theory and related topics. The chapters in this volume, compiled in his memory, are written by distinguished mathematicians and pay tribute to his many significant and lasting achievements.

This book provides an introduction to the broad topic of the calculus of variations. It addresses the most natural questions on variational problems and the mathematical complexities they present.Beginning with the scientific modeling that motivates the subject, the book then tackles mathematical questions such as the existence and uniqueness of solutions, their characterization in terms of partial differential equations, and their regularity. It includes both classical and recent results on one-dimensional variational problems, as well as the adaptation to the multi-dimensional case. Here, convexity plays an important role in establishing semi-continuity results and connections with techniques from optimization, and convex duality is even used to produce regularity results. This is then followed by the more classical Hölder regularity theory for elliptic PDEs and some geometric variational problems on sets, including the isoperimetric inequality andthe Steiner tree problem. The book concludes with a chapter on the limits of sequences of variational problems, expressed in terms of ¿-convergence.While primarily designed for master's-level and advanced courses, this textbook, based on its author's instructional experience, also offers original insights that may be of interest to PhD students and researchers. A foundational understanding of measure theory and functional analysis is required, but all the essential concepts are reiterated throughout the book using special memo-boxes.

This book presents fixed point theory, one of the crucial tools in applied mathematics, functional analysis, and topology, which has been used to solve distinct real-world problems in computer science, engineering, and physics. The authors begin with an overview of the extension of metric spaces. Readers are introduced to general fixed-point theorems while comparing and contrasting important and insignificant metric spaces. The book is intended to be self-contained and serves as a unique resource for researchers in various disciplines.

This book presents a curated selection of recent research in functional analysis and fixed-point theory, exploring their applications in interdisciplinary fields. The primary objective is to establish a connection between the latest developments in functional analysis and fixed-point theory and the broader interdisciplinary research landscape. By doing so, this book aims to address the needs of researchers and experts seeking to stay up-to-date with the cutting-edge research trends in functional analysis, fixed-point theory and related areas. It also aims to pave the way for applying functional analysis and fixed-point theory to solve interdisciplinary problems in various domains, including but not limited to fractional calculus, integral equations, queuing theory, convex analysis, harmonic analysis and wavelet analysis.

This book provides a valuable collection of contributions by distinguished scholars presenting the state of the art and some of the most significant latest developments and future challenges in the field of dispersive partial differential equations. The material covers four major lines: (1) Long time behaviour of NLS-type equations, (2) probabilistic and nonstandard methods in the study of NLS equation, (3) dispersive properties for heat-, Schrodinger-, and Dirac-type flows, (4) wave and KdV-type equations. Across a variety of applications an amount of crucial mathematical tools are discussed, whose applicability and versatility goes beyond the specific models presented here. Furthermore, all contributions include updated and comparative literature.

This monograph presents the solution of the classical moment problem, the construction of Jacobi matrices and corresponding polynomials. The cases of strongly,trigonometric, complex and real two-dimensional moment problems are discussed, and the Jacobi-type matrices corresponding to the trigonometric moment problem are shown. The Berezansky theory of the expansion in generalized eigenvectors for corresponding set of commuting operators plays the key role in the proof of results.The book is recommended for researchers in fields of functional analysis, operator theory, mathematical physics, and engineers who deal with problems of coupled pendulums.

This book, which is the first of two volumes, presents, in a unique way, some of the most relevant research tools of modern analysis. This work empowers young researchers with all the necessary techniques to explore the various subfields of this broad subject, and introduces relevant frameworks where these tools can be immediately deployed.Volume I starts with the foundations of modern analysis. The first three chapters are devoted to topology, measure theory, and functional analysis. Chapter 4 offers a comprehensive analysis of the main function spaces, while Chapter 5 covers more concrete subjects, like multivariate analysis, which are closely related to applications and more difficult to find in compact form. Chapter 6 deals with smooth and non-smooth calculus of functions; Chapter 7 introduces certain important classes of nonlinear operators; and Chapter 8 complements the previous three chapters with topics of variational analysis. Each chapter of this volume finishes with a list of problems - handy for understanding and self-study - and historical notes that give the reader a more vivid picture of how the theory developed. Volume II consists of various applications using the tools and techniques developed in this volume.By offering a clear and wide picture of the tools and applications of modern analysis, this work can be of great benefit not only to mature graduate students seeking topics for research, but also to experienced researchers with an interest in this vast and rich field of mathematics.

This volume examines current research in mechanics and its applications to various disciplines, with a particular focus on fluid-structure interaction (FSI). The topics have been chosen in commemoration of Dr. Bong Jae Chung and with respect to his wide range of research interests. This volume stands apart because of this diversity of interests, featuring an interdisciplinary and in-depth analysis of FSI that is difficult to find conveniently collected elsewhere in the literature. Contributors include mathematicians, physicists, mechanical and biomechanical engineers, and psychologists. This volume is structured into four thematic areas in order to increase its accessibility: theory, computations, experiments, and applications. Recent Advances in Mechanics and Fluid-Structure Interaction with Applications will appeal to established researchers as well as postdocs and graduate students interested in this active area of research.

This book explains the basic theory of Hilbert C*-module in detail, covering a wide range of applications from generalized index to module framework. At the center of the book, the Beurling-Deny criterion is characterized between operator valued Dirichlet forms and quantum Markov semigroups, hence opening a new field of quantum probability research. The general scope of the book includes: basic theory of Hilbert C*-modules; generalized indices and module frames; operator valued Dirichlet forms; and quantum Markov semigroups.This book will be of value to scholars and graduate students in the fields of operator algebra, quantum probability and quantum information.

This book discusses, develops and applies the theory of Vilenkin-Fourier series connected to modern harmonic analysis.The classical theory of Fourier series deals with decomposition of a function into sinusoidal waves. Unlike these continuous waves the Vilenkin (Walsh) functions are rectangular waves. Such waves have already been used frequently in the theory of signal transmission, multiplexing, filtering, image enhancement, code theory, digital signal processing and pattern recognition. The development of the theory of Vilenkin-Fourier series has been strongly influenced by the classical theory of trigonometric series. Because of this it is inevitable to compare results of Vilenkin-Fourier series to those on trigonometric series. There are many similarities between these theories, but there exist differences also. Much of these can be explained by modern abstract harmonic analysis, which studies orthonormal systems from the point of view of the structure of a topological group.The first part of the book can be used as an introduction to the subject, and the following chapters summarize the most recent research in this fascinating area and can be read independently. Each chapter concludes with historical remarks and open questions. The book will appeal to researchers working in Fourier and more broad harmonic analysis and will inspire them for their own and their students' research. Moreover, researchers in applied fields will appreciate it as a sourcebook far beyond the traditional mathematical domains.

This book focuses on the following three topics in the theory of boundary value problems of nonlinear second order elliptic partial differential equations and systems: (i) eigenvalue problem, (ii) upper and lower solutions method, (iii) topological degree method, and deals with the existence of solutions, more specifically non-constant positive solutions, as well as the uniqueness, stability and asymptotic behavior of such solutions. While not all-encompassing, these topics represent major approaches to the theory of partial differential equations and systems, and should be of significant interest to graduate students and researchers. Two appendices have been included to provide a good gauge of the prerequisites for this book and make it reasonably self-contained. A notable strength of the book is that it contains a large number of substantial examples. Exercises for the reader are also included. Therefore, this book is suitable as a textbook for graduate students who have already had an introductory course on PDE and some familiarity with functional analysis and nonlinear functional analysis, and as a reference for researchers.

This book publishes original research chapters on the theory of approximation by positive linear operators as well as theory of sequence spaces and illustrates their applications. Chapters are original and contributed by active researchers in the field of approximation theory and sequence spaces. Each chapter describes the problem of current importance and summarizes ways of their solution and possible applications which improve the current understanding pertaining to sequence spaces and approximation theory. The presentation of the articles is clear and self-contained throughout the book. 

Education plays a pivotal role in shaping the progress and development of any society. It is the foundation upon which civilizations are built and the key to unlocking a brighter future. In this subchapter, we will explore the significance of education in society, emphasizing the importance of cultivating a love for learning and igniting a passion for knowledge in everyone.Education is the catalyst that drives social, economic, and cultural advancements. It equips individuals with the necessary skills and knowledge to navigate the complexities of life, enabling them to contribute meaningfully to their communities. Education empowers individuals to make informed decisions, helps them understand the world around them, and encourages critical thinking. By promoting intellectual growth and self-discovery, education provides individuals with the tools to become active participants in society.A love for learning is the cornerstone of a well-rounded education. When individuals develop a genuine passion for acquiring knowledge, they become lifelong learners, continuously seeking to expand their minds and broaden their perspectives. This thirst for knowledge fuels personal growth and opens doors to endless opportunities. It fosters curiosity, creativity, and innovation, driving progress in all spheres of society.The benefits of education extend far beyond the individual. A society that values education thrives on the collective intelligence and expertise of its members. By investing in education, societies can address pressing issues, such as poverty, inequality, and environmental degradation. Education enables individuals to break free from the chains of ignorance and empowers them to create positive change.Moreover, education promotes social cohesion and harmony. It cultivates empathy and understanding, breaking down barriers and fostering respect for diversity. Education encourages individuals to embrace different cultures and perspectives, fostering a sense of global citizenship that is essential in today's interconnected world.

This book is a self-contained advanced monograph on inequalities involving the numerical radius of bounded linear operators acting on complex Hilbert spaces. The study of numerical range and numerical radius has a long and distinguished history starting from the Rayleigh quotients used in the 19th century to nowadays applications in quantum information theory and quantum computing.This monograph is intended for use by both researchers and graduate students of mathematics, physics, and engineering who have a basic background in functional analysis and operator theory. The book provides several challenging problems and detailed arguments for the majority of the results. Each chapter ends with some notes about historical views or further extensions of the topics. It contains a bibliography of about 180 items, so it can be used as a reference book including many classical and modern numerical radius inequalities.