This book gives a compact exposition of the fundamentals of the theory of locally convex topological vector spaces. Furthermore it contains a survey of the most important results of a more subtle nature, which cannot be regarded as basic, but knowledge which is useful for understanding applications. Finally, the book explores some of such applications connected with differential calculus and measure theory in infinite-dimensional spaces. These applications are a central aspect of the book, which is why it is different from the wide range of existing texts on topological vector spaces. Overall, this book develops differential and integral calculus on infinite-dimensional locally convex spaces by using methods and techniques of the theory of locally convex spaces. The target readership includes mathematicians and physicists whose research is related to infinite-dimensional analysis.

This volume is devoted to Joseph A. (Joe) Ball's contributions to operator theory and its applications and in celebration of his seventieth birthday. Joe Ball's career spans over four and a half decades, starting with his work on model theory and related topics for non-contractions and operators on multiply connected domains. Later on, more applied operator theory themes appeared in his work, involving factorization and interpolation for operator-valued functions, with extensive applications in system and control theory. He has worked on nonlinear control, time-varying systems and, more recently, on multidimensional systems and noncommutative H -theory on the unit ball and polydisk, and more general domains, and these are only the main themes in his vast oeuvre. Fourteen research papers constitute the core of this volume, written by mathematicians who have collaborated with Joe or have been influenced by his vast mathematical work. A curriculum vitae, a publications list and a list of Joe Ball's PhD students are included in this volume, as well as personal reminiscences by colleagues and friends. Contributions by Yu. M. Arlinskii, S. Hassi, M. Augat, J. W. Helton, I. Klep, S. McCullough, S. Balasubramanian, U. Wijesooriya, N. Cohen, Q. Fang, S. Gorai, J. Sarkar, G. J. Groenewald, S. ter Horst, J. Jaftha, A. C. M. Ran, M.A. Kaashoek, F. van Schagen, A. Kheifets, Z. A. Lykova, N. J. Young, A. E. Ajibo, R. T. W. Martin, A. Ramanantoanina, M.-J. Y. Ou, H. J. Woerdeman, A. van der Schaft, A. Tannenbaum, T. T. Georgiou, J. O. Deasy and L. Norton.

This book is devoted to conservative realizations of various classes of Stieltjes, inverse Stieltjes, and general Herglotz-Nevanlinna functions as impedance functions of linear systems. The main feature of the monograph is a new approach to the realization theory profoundly involving developed extension theory in triplets of rigged Hilbert spaces and unbounded operators as state-space operators of linear systems. The connections of the realization theory to systems with accretive, sectorial, and contractive state-space operators as well as to the Phillips-Kato sectorial extension problem, the Krein-von Neumann and Friedrichs extremal extensions are provided. Among other results the book contains applications to the inverse problems for linear systems with non-self-adjoint Schroedinger operators, Jacobi matrices, and to the Nevanlinna-Pick system interpolation.

Inequalities play a central role in mathematics with various applications in other disciplines. The main goal of this contributed volume is to present several important matrix, operator, and norm inequalities in a systematic and self-contained fashion. Some powerful methods are used to provide significant mathematical inequalities in functional analysis, operator theory and numerous fields in recent decades. Some chapters are devoted to giving a series of new characterizations of operator monotone functions and some others explore inequalities connected to log-majorization, relative operator entropy, and the Ando-Hiai inequality. Several chapters are focused on Birkhoff-James orthogonality and approximate orthogonality in Banach spaces and operator algebras such as C*-algebras from historical perspectives to current development. A comprehensive account of the boundedness, compactness, and restrictions of Toeplitz operators can be found in the book. Furthermore, an overview of the Bishop-Phelps-Bollobas theorem is provided. The state-of-the-art of Hardy-Littlewood inequalities in sequence spaces is given. The chapters are written in a reader-friendly style and can be read independently. Each chapter contains a rich bibliography. This book is intended for use by both researchers and graduate students of mathematics, physics, and engineering.

This book investigates the convergence and summability of both one-dimensional and multi-dimensional Fourier transforms, as well as the theory of Hardy spaces. To do so, it studies a general summability method known as theta-summation, which encompasses all the well-known summability methods, such as the Fejer, Riesz, Weierstrass, Abel, Picard, Bessel and Rogosinski summations. Following on the classic books by Bary (1964) and Zygmund (1968), this is the first book that considers strong summability introduced by current methodology. A further unique aspect is that the Lebesgue points are also studied in the theory of multi-dimensional summability. In addition to classical results, results from the past 20-30 years - normally only found in scattered research papers - are also gathered and discussed, offering readers a convenient "one-stop" source to support their work. As such, the book will be useful for researchers, graduate and postgraduate students alike.

This book gathers peer-reviewed contributions representing modern trends in the theory of generalized functions and pseudo-differential operators. It is dedicated to Professor Michael Oberguggenberger (Innsbruck University, Austria) in honour of his 60th birthday. The topics covered were suggested by the ISAAC Group in Generalized Functions (GF) and the ISAAC Group in Pseudo-Differential Operators (IGPDO), which met at the 9th ISAAC congress in Krakow, Poland in August 2013. Topics include Columbeau algebras, ultra-distributions, partial differential equations, micro-local analysis, harmonic analysis, global analysis, geometry, quantization, mathematical physics, and time-frequency analysis. Featuring both essays and research articles, the book will be of great interest to graduate students and researchers working in analysis, PDE and mathematical physics, while also offering a valuable complement to the volumes on this topic previously published in the OT series.

This second of two Exercises in Analysis volumes covers problems in five core topics of mathematical analysis: Function Spaces, Nonlinear and Multivalued Maps, Smooth and Nonsmooth Calculus, Degree Theory and Fixed Point Theory, and Variational and Topological Methods. Each of five topics corresponds to a different chapter with inclusion of the basic theory and accompanying main definitions and results,followed by suitable comments and remarks for better understanding of the material. Exercises/problems are presented for each topic, with solutions available at the end of each chapter. The entire collection of exercises offers a balanced and useful picture for the application surrounding each topic. This nearly encyclopedic coverage of exercises in mathematical analysis is the first of its kind and is accessible to a wide readership. Graduate students will find the collection of problems valuable in preparation for their preliminary or qualifying exams as well as for testing their deeper understanding of the material. Exercises are denoted by degree of difficulty. Instructors teaching courses that include one or all of the above-mentioned topics will find the exercises of great help in course preparation. Researchers in analysis may find this Work useful as a summary of analytic theories published in one accessible volume.

The theory of multivalued maps and the theory of differential inclusions are closely connected and intensively developing branches of contemporary mathematics. They have effective and interesting applications in control theory, optimization, calculus of variations, non-smooth and convex analysis, game theory, mathematical economics and in other fields.This book presents a user-friendly and self-contained introduction to both subjects. It is aimed at 'beginners', starting with students of senior courses. The book will be useful both for readers whose interests lie in the sphere of pure mathematics, as well as for those who are involved in applicable aspects of the theory. In Chapter 0, basic definitions and fundamental results in topology are collected. Chapter 1 begins with examples showing how naturally the idea of a multivalued map arises in diverse areas of mathematics, continues with the description of a variety of properties of multivalued maps and finishes with measurable multivalued functions. Chapter 2 is devoted to the theory of fixed points of multivalued maps. The whole of Chapter 3 focuses on the study of differential inclusions and their applications in control theory. The subject of last Chapter 4 is the applications in dynamical systems, game theory, and mathematical economics.The book is completed with the bibliographic commentaries and additions containing the exposition related both to the sections described in the book and to those which left outside its framework. The extensive bibliography (including more than 400 items) leads from basic works to recent studies.

This volume contains an important progress on the theory of subnormal operators in the past thirty years, which was developed by the author and his collaborators. It serves as a guide and basis to students and researchers on understanding and exploring further this new direction in operator theory. The volume expounds lucidly on analytic model theory, mosaics, trace formulas of the subnormal operators, and subnormal tuples of operators on the Hilbert spaces.

The chapters in this volume are based on talks given at the inaugural Aspects of Time-Frequency Analysis conference held in Turin, Italy from July 5-7, 2017, which brought together experts in harmonic analysis and its applications. New connections between different but related areas were presented in the context of time-frequency analysis, encouraging future research and collaborations. Some of the topics covered include: Abstract harmonic analysis, Numerical harmonic analysis, Sampling theory, Compressed sensing, Mathematical signal processing, Pseudodifferential operators, and Applications of harmonic analysis to quantum mechanics. Landscapes of Time-Frequency Analysis will be of particular interest to researchers and advanced students working in time-frequency analysis and other related areas of harmonic analysis.

Hilbert functions and resolutions are both central objects in commutative algebra and fruitful tools in the fields of algebraic geometry, combinatorics, commutative algebra, and computational algebra. Spurred by recent research in this area, Syzygies and Hilbert Functions explores fresh developments in the field as well as fundamental concepts. Written by international mathematics authorities, the book first examines the invariant of Castelnuovo-Mumford regularity, blowup algebras, and bigraded rings. It then outlines the current status of two challenging conjectures: the lex-plus-power (LPP) conjecture and the multiplicity conjecture. After reviewing results of the geometry of Hilbert functions, the book considers minimal free resolutions of integral subschemes and of equidimensional Cohen-Macaulay subschemes of small degree. It also discusses relations to subspace arrangements and the properties of the infinite graded minimal free resolution of the ground field over a projective toric ring. The volume closes with an introduction to multigraded Hilbert functions, mixed multiplicities, and joint reductions. By surveying exciting topics of vibrant current research, Syzygies and Hilbert Functions stimulates further study in this hot area of mathematical activity.

The aim of Summable Spaces and Their Duals, Matrix Transformations and Geometric Properties is to discuss primarily about different kinds of summable spaces, compute their duals and then characterize several matrix classes transforming one summable space into other. The book also discusses several geometric properties of summable spaces, as well as dealing with the construction of summable spaces using Orlicz functions, and explores several structural properties of such spaces. Each chapter contains a conclusion section highlighting the importance of results, and points the reader in the direction of possible new ideas for further study. Features Suitable for graduate schools, graduate students, researchers and faculty, and could be used as a key text for special Analysis seminars Investigates different types of summable spaces and computes their duals Characterizes several matrix classes transforming one summable space into other Discusses several geometric properties of summable spaces Examines several possible generalizations of Orlicz sequence spaces

The aim of Spectral Geometry of Partial Differential Operators is to provide a basic and self-contained introduction to the ideas underpinning spectral geometric inequalities arising in the theory of partial differential equations. Historically, one of the first inequalities of the spectral geometry was the minimization problem of the first eigenvalue of the Dirichlet Laplacian. Nowadays, this type of inequalities of spectral geometry have expanded to many other cases with number of applications in physics and other sciences. The main reason why the results are useful, beyond the intrinsic interest of geometric extremum problems, is that they produce a priori bounds for spectral invariants of (partial differential) operators on arbitrary domains. Features: Collects the ideas underpinning the inequalities of the spectral geometry, in both self-adjoint and non-self-adjoint operator theory, in a way accessible by anyone with a basic level of understanding of linear differential operators Aimed at theoretical as well as applied mathematicians, from a wide range of scientific fields, including acoustics, astronomy, MEMS, and other physical sciences Provides a step-by-step guide to the techniques of non-self-adjoint partial differential operators, and for the applications of such methods. Provides a self-contained coverage of the traditional and modern theories of linear partial differential operators, and does not require a previous background in operator theory.

The theory of Lebesgue and Sobolev spaces with variable integrability is experiencing a steady expansion, and is the subject of much vigorous research by functional analysts, function-space analysts and specialists in nonlinear analysis. These spaces have attracted attention not only because of their intrinsic mathematical importance as natural, interesting examples of non-rearrangement-invariant function spaces but also in view of their applications, which include the mathematical modeling of electrorheological fluids and image restoration.The main focus of this book is to provide a solid functional-analytic background for the study of differential operators on spaces with variable integrability. It includes some novel stability phenomena which the authors have recently discovered.At the present time, this is the only book which focuses systematically on differential operators on spaces with variable integrability. The authors present a concise, natural introduction to the basic material and steadily move toward differential operators on these spaces, leading the reader quickly to current research topics.

An important class of integral expansions generated by Sturm-Liouville theory involving spherical harmonics is commonly known as Mehler-Fock integral transforms. In this book, a number of integral expansions of such type have been established rigorously. As applications, integral expansions of some simple function are also obtained.

The general theories contained in the text will give rise to new ideas and methods for the natural inversion formulas for general linear mappings in the framework of Hilbert spaces containing the natural solutions for Fredholm integral equations of the first kind.

Presenting research from more than 30 international authorities, this reference provides a complete arsenal of tools and theorems to analyze systems of hyperbolic partial differential equations. The authors investigate a wide variety of problems in areas such as thermodynamics, electromagnetics, fluid dynamics, differential geometry, and topology. Renewing thought in the field of mathematical physics, Hyperbolic Differential Operators defines the notion of pseudosymmetry for matrix symbols of order zero as well as the notion of time function. Surpassing previously published material on the topic, this text is key for researchers and mathematicians specializing in hyperbolic, SchrAdinger, Einstein, and partial differential equations; complex analysis; and mathematical physics.

Multifractal theory was introduced by theoretical physicists in 1986. Since then, multifractals have increasingly been studied by mathematicians. This new work presents the latest research on random results on random multifractals and the physical thermodynamical interpretation of these results. As the amount of work in this area increases, Lars Olsen presents a unifying approach to current multifractal theory. Featuring high quality, original research material, this important new book fills a gap in the current literature available, providing a rigorous mathematical treatment of multifractal measures.

Specialized as it might be, continuum theory is one of the most intriguing areas in mathematics. However, despite being popular journal fare, few books have thoroughly explored this interesting aspect of topology. In Topics on Continua, Sergio Macias, one of the field's leading scholars, presents four of his favorite continuum topics: inverse limits, Jones's set function T, homogenous continua, and n-fold hyperspaces, and in doing so, presents the most complete set of theorems and proofs ever contained in a single topology volume. Many of the results presented have previously appeared only in research papers, and some appear here for the first time. After building the requisite background and exploring the inverse limits of continua, the discussions focus on Professor Jones's set function T and continua for which T is continuous. An introduction to topological groups and group actions lead to a proof of Effros's Theorem, followed by a presentation of two decomposition theorems. The author then offers an in-depth study of n-fold hyperspaces. This includes their general properties, conditions that allow points of n-fold symmetric products to be arcwise accessible from their complement, points that arcwise disconnect the n-fold hyperspaces, the n-fold hyperspaces of graphs, and theorems relating n-fold hyperspaces and cones. The concluding chapter presents a series of open questions on each topic discussed in the book. With more than a decade of teaching experience, Macias is able to put forth exceptionally cogent discussions that not only give beginning mathematicians a strong grounding in continuum theory, but also form an authoritative, single-source guide through some of topology's most captivating facets.

Continuing the strong tradition of functional analysis and stability theory for differential and integral equations already established by the previous volumes in this series, this innovative monograph considers in detail the method of limiting equations constructed in terms of the Bebutov-Miller-Sell concept, the method of comparison, and Lyapunov's direct method based on scalar, vector and matrix functions. The stability of abstract compacted and uniform dynamic processes, dispersed systems and evolutionary equations in Banach space are also discussed. For the first time, the method first employed by Krylov and Bogolubov in their investigations of oscillations in almost linear systems is applied to a new field: that of the stability problem of systems with small parameters. This important development should facilitate the solution of engineering problems in such areas as orbiting satellites, rocket motion, high-speed vehicles, power grids, and nuclear reactors.

Along with more than 2100 integral equations and their solutions, this handbook outlines exact analytical methods for solving linear and nonlinear integral equations and provides an evaluation of approximate methods. Each section provides examples that show how methods can be applied to specific equations.

This volume presents a systematic and unified treatment of Leray-Schauder continuation theorems in nonlinear analysis. In particular, fixed point theory is established for many classes of maps, such as contractive, non-expansive, accretive, and compact maps, to name but a few. This book also presents coincidence and multiplicity results. Many applications of current interest in the theory of nonlinear differential equations are presented to complement the theory. The text is essentially self-contained, so it may also be used as an introduction to topological methods in nonlinear analysis. This volume will appeal to graduate students and researchers in mathematical analysis and its applications.

This book gives an excellent and up-to-date overview on the convergence and joint progress in the fields of Generalized Functions and Fourier Analysis, notably in the core disciplines of pseudodifferential operators, microlocal analysis and time-frequency analysis. The volume is a collection of chapters addressing these fields, their interaction, their unifying concepts and their applications and is based on scientific activities related to the International Association for Generalized Functions (IAGF) and the ISAAC interest groups on Pseudo-Differential Operators (IGPDO) and on Generalized Functions (IGGF), notably on the longstanding collaboration of these groups within ISAAC.

Until now, no book has systematically presented the recently developed concept of envelopes in function spaces. Envelopes are relatively simple tools for the study of classical and more complicated spaces, such as Besov and Triebel-Lizorkin types, in limiting situations. This theory originates from the classical result of the Sobolev embedding theorem, ubiquitous in all areas of functional analysis. Self-contained and accessible, Envelopes and Sharp Embeddings of Function Spaces provides the first detailed account of the new theory of growth and continuity envelopes in function spaces. The book is well structured into two parts, first providing a comprehensive introduction and then examining more advanced topics. Some of the classical function spaces discussed in the first part include Lebesgue, Lorentz, Lipschitz, and Sobolev. The author defines growth and continuity envelopes and examines their properties. In Part II, the book explores the results for function spaces of Besov and Triebel-Lizorkin types. The author then presents several applications of the results, including Hardy-type inequalities, asymptotic estimates for entropy, and approximation numbers of compact embeddings. As one of the key researchers in this progressing field, the author offers a coherent presentation of the recent developments in function spaces, providing valuable information for graduate students and researchers in functional analysis.

The seminal 1989 work of Douglas and Paulsen on the theory of Hilbert modules over function algebras precipitated a number of major research efforts. This in turn led to some intriguing and valuable results, particularly in the areas of operator theory and functional analysis. With the field now beginning to blossom, the time has come to collect those results in one volume.

Written by two of the most active and often-cited researchers in the field, Analytic Hilbert Modules offers a clear, logical survey of recent developments, including advances made by authors and others. It provides much-needed insight into function theory of several variables and includes significant results published here for the first time in areas such as characteristic space theory, rigidity phenomena, the equivalence problem, Arveson modules, extension theory, and reproducing Hilbert spaces on n-dimensional complex space.