Functional analysis and transforms Books

Book SynopsisModern Mathematical Methods for Scientists and Engineers is a modern introduction to basic topics in mathematics at the undergraduate level, with emphasis on explanations and applications to real-life problems. There is also an 'Application' section at the end of each chapter, with topics drawn from a variety of areas, including neural networks, fluid dynamics, and the behavior of 'put' and 'call' options in financial markets. The book presents several modern important and computationally efficient topics, including feedforward neural networks, wavelets, generalized functions, stochastic optimization methods, and numerical methods.A unique and novel feature of the book is the introduction of a recently developed method for solving partial differential equations (PDEs), called the unified transform. PDEs are the mathematical cornerstone for describing an astonishingly wide range of phenomena, from quantum mechanics to ocean waves, to the diffusion of heat in matter and the behavior of financial markets. Despite the efforts of many famous mathematicians, physicists and engineers, the solution of partial differential equations remains a challenge.The unified transform greatly facilitates this task. For example, two and a half centuries after Jean d'Alembert formulated the wave equation and presented a solution for solving a simple problem for this equation, the unified transform derives in a simple manner a generalization of the d'Alembert solution, valid for general boundary value problems. Moreover, two centuries after Joseph Fourier introduced the classical tool of the Fourier series for solving the heat equation, the unified transform constructs a new solution to this ubiquitous PDE, with important analytical and numerical advantages in comparison to the classical solutions. The authors present the unified transform pedagogically, building all the necessary background, including functions of real and of complex variables and the Fourier transform, illustrating the method with numerous examples.Broad in scope, but pedagogical in style and content, the book is an introduction to powerful mathematical concepts and modern tools for students in science and engineering.

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Book SynopsisIntended for students with a beginning knowledge of mathematical analysis, this first volume, in a three-part introduction to Fourier analysis, introduces the core areas of mathematical analysis while also illustrating the organic unity between them. It includes numerous examples and applications.Table of ContentsForeword vii Preface xi Chapter 1. The Genesis of Fourier Analysis 1 Chapter 2. Basic Properties of Fourier Series 29 Chapter 3. Convergence of Fourier Series 69 Chapter 4. Some Applications of Fourier Series 100 Chapter 5. The Fourier Transform on R 129 Chapter 6. The Fourier Transform on R d 175 Chapter 7. Finite Fourier Analysis 218 Chapter 8. Dirichlet's Theorem 241 Appendix: Integration 281 Notes and References 299 Bibliography 301 Symbol Glossary 305

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Book SynopsisConfusing Textbooks? Missed Lectures? Not Enough Time?Fortunately for you, there's Schaum's Outlines. More than 40 million students have trusted Schaum's to help them succeed in the classroom and on exams. Schaum's is the key to faster learning and higher grades in every subject. Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. You also get hundreds of examples, solved problems, and practice exercises to test your skills. This Schaum's Outline gives you Practice problems with full explanations that reinforce knowledge Coverage of the most up-to-date developments in your course field In-depth review of practices and applications Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten your study time-and get your best test scores!Schaum's Outlines-Problem Solved.Table of ContentsThe Laplace Transform.The Inverse Laplace Transform.Applications to Differential Equations.Applications to Integral and Difference Equations.Complex Variable Theory.Fourier Series and Integrals.The Complex Inversion Formula.Applications to Boundary-Value Problems.Appendix A: Table of General Properties of Laplace Transforms.Appendix B: Table of Special Laplace Transforms.Appendix C: Table of Special Functions.

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Book SynopsisD'oh! Fourier introduces the Fourier transform and is aimed at undergraduates in Computer Science, Mathematics, and Applied Sciences, as well as for those wishing to extend their education. Formulated around ten key points, this accessible book is light-hearted and illustrative, with many applications. The basis and deployment of the Fourier transform are covered applying real-world examples throughout inductively rather than the theoretical approach deductively.The key components of the textbook are continuous signals analysis, discrete signals analysis, image processing, applications of Fourier analysis, together with the origin and nature of the transform itself. D'oh! Fourier is reproducible via MATLAB/Octave and is supported by a comprehensive website which provides the code contained within the book.

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Book SynopsisThe focus of this book is on open conformal dynamical systems corresponding to the escape of a point through an open Euclidean ball. The ultimate goal is to understand the asymptotic behavior of the escape rate as the radius of the ball tends to zero. In the case of hyperbolic conformal systems this has been addressed by various authors. The conformal maps considered in this book are far more general, and the analysis correspondingly more involved. The asymptotic existence of escape rates is proved and they are calculated in the context of (finite or infinite) countable alphabets, uniformly contracting conformal graph-directed Markov systems, and in particular, conformal countable alphabet iterated function systems. These results have direct applications to interval maps, rational functions and meromorphic maps. Towards this goal the authors develop, on a purely symbolic level, a theory of singular perturbations of Perron--Frobenius (transfer) operators associated with countable alphabet subshifts of finite type and Hölder continuous summable potentials. This leads to a fairly full account of the structure of the corresponding open dynamical systems and their associated surviving sets.Table of Contents1. Introduction.- 2. Singular Perturbations of Classical Original Perron–Frobenius Operators on Countable Alphabet Symbol Spaces.- 3. Symbol Escape Rates and the Survivor Set K(Un).- 4. Escape Rates for Conformal GDMSs and IFSs.- 5. Applications: Escape Rates for Multimodal Mapsand One-Dimensional Complex Dynamics.

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Book Synopsis"Chemists familiar with conventional quantum mechanics will applaud and benefit greatly from this particularly instructive, thorough and clearly written exposition of density functional theory: its basis, concepts, terms, implementation, and performance in diverse applications. Users of DFT for structure, energy, and molecular property computations, as well as reaction mechanism studies, are guided to the optimum choices of the most effective methods. Well done!" Paul von Rague Schleyer "A conspicuous hole in the computational chemist's library is nicely filled by this book, which provides a wide-ranging and pragmatic view of the subject.[...It] should justifiably become the favorite text on the subject for practioneers who aim to use DFT to solve chemical problems." J. F. Stanton, J. Am. Chem. Soc. "The authors' aim is to guide the chemist through basic theoretical and related technical aspects of DFT at an easy-to-understand theoretical level. They succeed admirably." P. C. H. Mitchell, Appl. Organomet. Chem. "The authors have done an excellent service to the chemical community. [...] A Chemist's Guide to Density Functional Theory is exactly what the title suggests. It should be an invaluable source of insight and knowledge for many chemists using DFT approaches to solve chemical problems." M. Kaupp, Angew. Chem. Table of ContentsPART A: THE DEFINITION OF THE MODEL Elementary Quantum Chemistry Electron Density and Hole Functions The Electron Density as Basic Variable: Early Attempts The Hohenberg-Kohn Theorems The Kohn-Sham Approach The Quest for Approximate Exchange-Correlation Functionals The Basic Machinery of Density Functional Programs PART B: THE PERFORMANCE OF THE MODEL Molecular Structures and Vibrational Frequencies Relative Energies and Thermochemistry Electric Properties Magnetic Properties Hydrogen Bonds and Weakly Bound Systems Chemical Reactivity: Exploration of Potential Energy Surfaces

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Book SynopsisThis text is meant to be a hands-on lab manual that can be used in class every day to guide the exploration of the theory and applications of differential and integral calculus. For the most part, labs can be used individually or in a sequence. Each lab consists of an explanation of material with integrated exercises. Some labs are split into multiple subsections and thus exercises are separated by those subsections. The exercise sections integrate problems, technology, Mathematica R visualization, and Mathematica CDFs that allow students to discover the theory and applications of differential and integral calculus in a meaningful and memorable way.Employs Mathematica to calculate and explore concepts and theories of calculusUses engaging labs to inspire learningIncludes many applications to a variety of fields that can promote research projectsUser-friendly approach that can be used for classroom work or independent exploratory leaTable of ContentsLimits and Continuity. Derivatives and Their Applications. Areas, Integrals, and Accumulation. Applications of Antiderivatives. Mathematica Demonstrations and References.

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Book SynopsisThis brief monograph on the gamma function by a major 20th century mathematician was designed to bridge a gap in the literature of mathematics between incomplete and over-complicated treatments. Topics include functions, the Euler integrals and the Gauss formula, large values of x and the multiplication formula, the connection with sin x, applications to definite integrals, and other subjects.

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Book SynopsisNow revised and updated, this brisk introduction to functional analysis is intended for advanced undergraduate students, typically final year, who have had some background in real analysis. The author's aim is not just to cover the standard material in a standard way, but to present results of application in contemporary mathematics and to show the relevance of functional analysis to other areas. Unusual topics covered include the geometry of finite-dimensional spaces, invariant subspaces, fixed-point theorems, and the Bishop-Phelps theorem. An outstanding feature is the large number of exercises, some straightforward, some challenging, none uninteresting.Trade Review' … a well-written concise introduction to functional analysis.' European Mathematical Society'Bollobás writes with clarity and has clearly thought about the needs of his readers. First-time students of functional analysis will thank him for his willingness to remind them about notation and to repeat definitions that he has not used for a while. Bollobás has written a fine book. it is an excellent introduction to functional analysis that will be invaluable to advanced undergraduate students (and their lectures). Steve Abbott, The Mathematical GazetteTable of ContentsPreface; 1. Basic inequalities; 2. Normed spaces and bounded linear operators; 3. Linear functional and the Hahn-Banach theorem; 4. Finite-dimensional normed spaces; 5. The Baire category theorem and the closed-graph theorem; 6. Continuous functions on compact spaces and the Stone-Weierstrass theorem; 7. The contraction-mapping theorem; 8. Weak topologies and duality; 9. Euclidean spaces and Hilbert spaces; 10. Orthonormal systems; 11. Adjoint operators; 12. The algebra of bounded linear operators; 13. Compact operators on Banach spaces; 14. Compact normal operators; 15. Fixed-point theorems; 16. Invariant subspaces; Index of notation; Index of terms.

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Book SynopsisThe purpose of this monograph is to offer an accessible and essentially self-contained presentation of some mathematical aspects of the Feynman path integral in non-relativistic quantum mechanics. In spite of the primary role in the advancement of modern theoretical physics and the wide range of applications, path integrals are still a source of challenging problem for mathematicians. From this viewpoint, path integrals can be roughly described in terms of approximation formulas for an operator (usually the propagator of a Schrödinger-type evolution equation) involving a suitably designed sequence of operators.In keeping with the spirit of harmonic analysis, the guiding theme of the book is to illustrate how the powerful techniques of time-frequency analysis - based on the decomposition of functions and operators in terms of the so-called Gabor wave packets – can be successfully applied to mathematical path integrals, leading to remarkable results and paving the way to a fruitful interaction.This monograph intends to build a bridge between the communities of people working in time-frequency analysis and mathematical/theoretical physics, and to provide an exposition of the present novel approach along with its basic toolkit. Having in mind a researcher or a Ph.D. student as reader, we collected in Part I the necessary background, in the most suitable form for our purposes, following a smooth pedagogical pattern. Then Part II covers the analysis of path integrals, reflecting the topics addressed in the research activity of the authors in the last years.Table of Contents- Itinerary - How Gabor Analysis met Feynman Path Integrals. - Part I Elements of Gabor Analysis. - Basic Facts of Classical Analysis. - The Gabor Analysis of Functions. - The Gabor Analysis of Operators. - Semiclassical Gabor Analysis. - Part II Analysis of Feynman Path Integrals. - Pointwise Convergence of the Integral Kernels. - Convergence in L(L2) - Potentials in the Sjöstrand Class. - Convergence in L(L2) - Potentials in Kato-Sobolev Spaces. - Convergence in the Lp Setting.

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Book Synopsis​The book The E. M. Stein Lectures on Hardy Spaces is based on a graduate course on real variable Hardy spaces which was given by E.M. Stein at Princeton University in the academic year 1973-1974. Stein, along with C. Fefferman and G. Weiss, pioneered this subject area, removing the theory of Hardy spaces from its traditional dependence on complex variables, and to reveal its real-variable underpinnings.This book is based on Steven G. Krantz’s notes from the course given by Stein. The text builds on Fefferman's theorem that BMO is the dual of the Hardy space. Using maximal functions, singular integrals, and related ideas, Stein offers many new characterizations of the Hardy spaces. The result is a rich tapestry of ideas that develops the theory of singular integrals to a new level. The final chapter describes the major developments since 1974.This monograph is of broad interest to graduate students and researchers in mathematical analysis. Prerequisites for the book include a solid understanding of real variable theory and complex variable theory. A basic knowledge of functional analysis would also be useful.Table of ContentsIntroductory material.- More on Hardy Spaces.- Background on H^p Spaces.- Hardy Spaces on D.- Hardy Spaces on R^n.- Developments Since 1974.- Concluding Remarks.- Bibliography.- Index.

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Book SynopsisThis book studies the modules arising in Fourier expansions of automorphic forms, namely Fourier term modules on SU(2,1), the smallest rank one Lie group with a non-abelian unipotent subgroup. It considers the “abelian” Fourier term modules connected to characters of the maximal unipotent subgroups of SU(2,1), and also the “non-abelian” modules, described via theta functions. A complete description of the submodule structure of all Fourier term modules is given, with a discussion of the consequences for Fourier expansions of automorphic forms, automorphic forms with exponential growth included.These results can be applied to prove a completeness result for Poincaré series in spaces of square integrable automorphic forms.Aimed at researchers and graduate students interested in automorphic forms, harmonic analysis on Lie groups, and number-theoretic topics related to Poincaré series, the book will also serve as a basic reference on spectral expansion with Fourier-Jacobi coefficients. Only a background in Lie groups and their representations is assumed.Table of Contents- 1. Introduction. - 2. The Lie Group SU(2,1) and Subgroups. - 3. Fourier Term Modules. - 4. Submodule Structure. - 5. Application to Automorphic Forms.

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Book Synopsis- Part I Quaternionic functional analysis and related topics.- Functions of a Quaternionic Variable.- Quaternionic Banach and Hilbert Spaces.- Quaternionic Linear Operators on a Hilbert Space.- Reproducing Kernel Hilbert Spaces.- Part II Spaces of Slice Hyperholomorphic Functions.- Slice Hyperholomorphic Hardy Spaces.- de Branges Spaces.- Slice Hyperholomorphic Bergman Spaces.- Slice Hyperholomorphic Bloch, Besov and Dirichlet Spaces.- Slice Hyperholomorphic Fock Space.- Wiener Algebras.- Part III Various Extensions.- Slice Polyanalytic Functions.- Several Quaternionic Variables.- Function Spaces and Spectral Theories.

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Book SynopsisThis book offers a systematic introduction to the optimal stochastic control theory via the dynamic programming principle, which is a powerful tool to analyze control problems.First we consider completely observable control problems with finite horizons. Using a time discretization we construct a nonlinear semigroup related to the dynamic programming principle (DPP), whose generator provides the Hamilton–Jacobi–Bellman (HJB) equation, and we characterize the value function via the nonlinear semigroup, besides the viscosity solution theory. When we control not only the dynamics of a system but also the terminal time of its evolution, control-stopping problems arise. This problem is treated in the same frameworks, via the nonlinear semigroup. Its results are applicable to the American option price problem.Zero-sum two-player time-homogeneous stochastic differential games and viscosity solutions of the Isaacs equations arising from such games are studied via a nonlinear semigroup related to DPP (the min-max principle, to be precise). Using semi-discretization arguments, we construct the nonlinear semigroups whose generators provide lower and upper Isaacs equations.Concerning partially observable control problems, we refer to stochastic parabolic equations driven by colored Wiener noises, in particular, the Zakai equation. The existence and uniqueness of solutions and regularities as well as Itô's formula are stated. A control problem for the Zakai equations has a nonlinear semigroup whose generator provides the HJB equation on a Banach space. The value function turns out to be a unique viscosity solution for the HJB equation under mild conditions.This edition provides a more generalized treatment of the topic than does the earlier book Lectures on Stochastic Control Theory (ISI Lecture Notes 9), where time-homogeneous cases are dealt with. Here, for finite time-horizon control problems, DPP was formulated as a one-parameter nonlinear semigroup, whose generator provides the HJB equation, by using a time-discretization method. The semigroup corresponds to the value function and is characterized as the envelope of Markovian transition semigroups of responses for constant control processes. Besides finite time-horizon controls, the book discusses control-stopping problems in the same frameworks.

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Book SynopsisThis book provides a concise and meticulous introduction to functional analysis. Since the topic draws heavily on the interplay between the algebraic structure of a linear space and the distance structure of a metric space, functional analysis is increasingly gaining the attention of not only mathematicians but also scientists and engineers. The purpose of the text is to present the basic aspects of functional analysis to this varied audience, keeping in mind the considerations of applicability. A novelty of this book is the inclusion of a result by Zabreiko, which states that every countably subadditive seminorm on a Banach space is continuous. Several major theorems in functional analysis are easy consequences of this result.The entire book can be used as a textbook for an introductory course in functional analysis without having to make any specific selection from the topics presented here. Basic notions in the setting of a metric space are defined in terms of sequences. These include total boundedness, compactness, continuity and uniform continuity. Offering concise and to-the-point treatment of each topic in the framework of a normed space and of an inner product space, the book represents a valuable resource for advanced undergraduate students in mathematics, and will also appeal to graduate students and faculty in the natural sciences and engineering. The book is accessible to anyone who is familiar with linear algebra and real analysis.Trade Review“The title of this book indicates that it is mainly devoted to linear maps on linear spaces. … All chapters are accompanied by useful exercises of varying levels of difficulty, which help the readers to develop their knowledge on the topics. The solutions of the exercises are given at the end of the book. … This textbook is essentially addressed to people working in engineering and sciences branches.” (Mohammad Sal Moslehian, zbMATH 1352.46001, 2017)Table of ContentsChapter 1. Preliminaries.- Chapter 2. Basic Framework.- Chapter 3. Bounded Linear Maps.- Chapter 4. Dual Spaces, Transposes and Adjoints.- Chapter 5. Spectral Theory.

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Book SynopsisThis book exposes the internal structure of non-self-adjoint operators acting on complex separable infinite dimensional Hilbert space, by analyzing and studying the commutant of operators. A unique presentation of the theorem of Cowen-Douglas operators is given. The authors take the strongly irreducible operator as a basic model, and find complete similarity invariants of Cowen-Douglas operators by using K-theory, complex geometry and operator algebra tools.

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Book SynopsisThis book is devoted to norm estimates for operator-valued functions of one and two operator arguments, as well as to their applications to spectrum perturbations of operators and to linear operator equations, i.e. to equations whose solutions are linear operators. Linear operator equations arise in both mathematical theory and engineering practice. The norm estimates suggested in the book have applications to the theories of ordinary differential, difference, functional-differential and integro-differential equations, as well as to the theories of integral operators and analytic functions. This book provides new tools for specialists in matrix theory and functional analysis. A significant part of the book covers the theory of triangular representations of operators that was developed by L de Branges, M S Brodskii, I C Gohberg, M G Krein, M S Livsic and other mathematicians.

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Book Synopsis?? Provides a concise but rigorous account of the theoretical background of FDA. ?? Introduces topics in various areas of mathematics, probability and statistics from the perspective of FDA. ?? Presents a systematic exposition of the fundamental statistical issues in FDA.Table of ContentsPreface xi 1 Introduction 1 1.1 Multivariate analysis in a nutshell 2 1.2 The path that lies ahead 13 2 Vector and function spaces 15 2.1 Metric spaces 16 2.2 Vector and normed spaces 20 2.3 Banach and Lp spaces 26 2.4 Inner Product and Hilbert spaces 31 2.5 The projection theorem and orthogonal decomposition 38 2.6 Vector integrals 40 2.7 Reproducing kernel Hilbert spaces 46 2.8 Sobolev spaces 55 3 Linear operator and functionals 61 3.1 Operators 62 3.2 Linear functionals 66 3.3 Adjoint operator 71 3.4 Nonnegative, square-root, and projection operators 74 3.5 Operator inverses 77 3.6 Fréchet and Gâteaux derivatives 83 3.7 Generalized Gram–Schmidt decompositions 87 4 Compact operators and singular value decomposition 91 4.1 Compact operators 92 4.2 Eigenvalues of compact operators 96 4.3 The singular value decomposition 103 4.4 Hilbert–Schmidt operators 107 4.5 Trace class operators 113 4.6 Integral operators and Mercer’s Theorem 116 4.7 Operators on an RKHS 123 4.8 Simultaneous diagonalization of two nonnegative definite operators 126 5 Perturbation theory 129 5.1 Perturbation of self-adjoint compact operators 129 5.2 Perturbation of general compact operators 140 6 Smoothing and regularization 147 6.1 Functional linear model 147 6.2 Penalized least squares estimators 150 6.3 Bias and variance 157 6.4 A computational formula 158 6.5 Regularization parameter selection 161 6.6 Splines 165 7 Random elements in a Hilbert space 175 7.1 Probability measures on a Hilbert space 176 7.2 Mean and covariance of a random element of a Hilbert space 178 7.3 Mean-square continuous processes and the Karhunen–Lòeve Theorem 184 7.4 Mean-square continuous processes in L2 (E,B(E), mu) 190 7.5 RKHS valued processes 195 7.6 The closed span of a process 198 7.7 Large sample theory 203 8 Mean and covariance estimation 211 8.1 Sample mean and covariance operator 212 8.2 Local linear estimation 214 8.3 Penalized least-squares estimation 231 9 Principal components analysis 251 9.1 Estimation via the sample covariance operator 253 9.2 Estimation via local linear smoothing 255 9.3 Estimation via penalized least squares 261 10 Canonical correlation analysis 265 10.1 CCA for random elements of a Hilbert space 267 10.2 Estimation 274 10.3 Prediction and regression 281 10.4 Factor analysis 284 10.5 MANOVA and discriminant analysis 288 10.6 Orthogonal subspaces and partial cca 294 11 Regression 305 11.1 A functional regression model 305 11.2 Asymptotic theory 308 11.3 Minimax optimality 318 11.4 Discretely sampled data 321 References 327 Index 331 Notation Index 334

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Book SynopsisMost existing books on wavelets are either too mathematical or they focus on too narrow a specialty. This book provides a thorough treatment of the subject from an engineering point of view. It is a one-stop source of theory, algorithms, applications, and computer codes related to wavelets. This second edition has been updated by the addition of: a section on Other Wavelets that describes curvelets, ridgelets, lifting wavelets, etc a section on lifting algorithms Sections on Edge Detection and Geophysical Applications Section on Multiresolution Time Domain Method (MRTD) and on Inverse problems Trade Review"This book provides a thorough treatment of wavelet theory and is very convenient for graduate students and researchers in electrical engineering, physics, and applied mathematics." (Zentralblatt MATH, 2011) Table of Contents1.What is this book all about? 2. Mathematical Preliminary. 2.1 Linear Spaces. 2.2 Vectors and Vector Spaces. 2.3 Basis Functions, Orthogonality and Biothogonality. 2.4 Local Basis and Riesz Basis. 2.5 Discrete Linear Normed Space. 2.6 Approximation by Orthogonal Projection. 2.7 Matrix Algebra and Linear Transformation. 2.8 Digital Signals. 2.9 Exercises. 2.10 References. 3. Fourier Analysis. 3.1 Fourier Series. 3.2 Rectified Sine Wave. 3.3 Fourier Transform. 3.4 Properties of Fourier Transform. 3.5 Examples of Fourier Transform. 3.6 Poisson’s Sum and Partition of ZUnity. 3.7 Sampling Theorem. 3.8 Partial Sum and Gibb’s Phenomenon. 3.9 Fourier Analysis of Discrete-Time Signals. 3.10 Discrete Fourier Transform (DFT). 3.11 Exercise. 3.12 References. 4. Time-Frequency Analysis. 4.1 Window Function. 4.2 Short-Time Fourier Transform. 4.3 Discrete Short-Time Fourier Transform. 4.4 Discrete Gabor Representation. 4.5 Continuous Wavelet Transform. 4.6 Discrete Wavelet Transform. 4.7 Wavelet Series. 4.8 Interpretations of the Time-Frequency Plot. 4.9 Wigner-Ville Distribution. 4.10 Properties of Wigner-Ville Distribution. 4.11 Quadratic Superposition Principle. 4.12 Ambiguity Function. 4.13 Exercise. 4.14 Computer Programs. 4.15 References. 5. Multiresolution Anaylsis. 5.1 Multiresolution Spaces. 5.2 Orthogonal, Biothogonal, and Semiorthogonal Decomposition. 5.3 Two-Scale Relations. 5.4 Decomposition Relation. 5.5 Spline Functions and Properties. 5.6 Mapping a Function into MRA Space. 5.7 Exercise. 5.8 Computer Programs. 5.9 References. 6. Construction of Wavelets. 6.1 Necessary Ingredients for Wavelet Construction. 6.2 Construction of Semiorthogonal Spline Wavelets. 6.3 Construction of Orthonormal Wavelets. 6.4 Orthonormal Scaling Functions. 6.5 Construction of Biothogonal Wavelets. 6.6 Graphical Display of Wavelet. 6.7 Exercise. 6.8 Computer Programs. 6.9 References. 7. DWT and Filter Bank Algorithms. 7.1 Decimation and Interpolation. 7.2 Signal Representation in the Approximation Subspace. 7.3 Wavelet Decomposition Algorithm. 7.4 Reconstruction Algorithm. 7.5 Change of Bases. 7.6 Signal Reconstruction in Semiorthogonal Subspaces. 7.7 Examples. 7.8 Two-Channel Perfect Reconstruction Filter Bank. 7.9 Polyphase Representation for Filter Banks. 7.10 Comments on DWT and PR Filter Banks. 7.11 Exercise. 7.12 Computer Program. 7.13 References. 8. Special Topics in Wavelets and Algorithms. 8.1 Fast Integral Wavelet Transform. 8.2 Ridgelet Transform. 8.3 Curvelet Transform. 8.4 Complex Wavelets. 8.5 Lifting Wavelet transform. 8.6 References. 9. Digital Signal Processing Applications. 9.1 Wavelet Packet. 9.2 Wavelet-Packet Algorithms. 9.3 Thresholding. 9.4 Interference Suppression. 9.5 Faulty Bearing Signature Identification. 9.6 Two-Dimensional Wavelets and Wavelet Packets. 9.7 Edge Detection. 9.8 Image Compression. 9.9 Microcalcification Cluster Detection.

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Book SynopsisThis book provides a modern and up-to-date treatment of the Hilberttransform of distributions and the space of periodic distributions.Taking a simple and effective approach to a complex subject, thisvolume is a first-rate textbook at the graduate level as well as anextremely useful reference for mathematicians, applied scientists,and engineers. The author, a leading authority in the field, shares with thereader many new results from his exhaustive research on the Hilberttransform of Schwartz distributions. He describes in detail how touse the Hilbert transform to solve theoretical and physicalproblems in a wide range of disciplines; these include aerofoilproblems, dispersion relations, high-energy physics, potentialtheory problems, and others. Innovative at every step, J. N. Pandey provides a new definitionfor the Hilbert transform of periodic functions, which isespecially useful for those working in the area of signalprocessing for computational purposes. This definiTable of ContentsThe Riemann-Hilbert Problem. The Hilbert Transform of Distributions in D'Lp, 1 p infinity. The Hilbert Transform of Schwartz Distributions. n-Dimensional Hilbert Transform. Further Applications of the Hilbert Transform, the HilbertProblem-- A Distributional Approach. Periodic Distributions, Their Hilbert Transform andApplications. Bibliography. Indexes.

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Book SynopsisThis up-to-date account brings together results previously scattered throughout the literature as well as new material in the area of function theory. The focus is on describing some of what has been learned thus far about the structure of the de Branges-Rovnyak spaces and their function-theoretic connections.Table of ContentsHilbert Spaces Inside Hilbert Spaces. Hilbert Spaces Inside H?. Cauchy Integral Representations. Nonextreme Points. Extreme Points. Angular Derivatives. Higher Derivatives. Equality of H(b) and H(). Equality of H(b) and M(a). Near Equality of H(b) and M(a). Brief Mention of a Few Additional Topics. References. Supplementary References. Index.

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Book SynopsisFourier Series and Optical Transform Techniques in ContemporaryOptics An Introduction For anyone new to Fourier methods, this remarkable book willilluminate the subject like no other currently available. With over280 illustrations generated by computer graphics, it depicts in3-space (rather than the usual 2-space) the many basic functions ofoptical diffraction and imaging. These mind-stretchingvisualizations give the reader an enhanced understanding of bothFourier transform techniques and key principles in optics. At thesame time, the author provides a lucid text that covers wavenotation, the Fourier analysis of signals, the processing of lightin diffraction phenomena and imaging, Zernicke polynomials, Fouriertransforms for Fresnel diffraction, laser beacon adaptive optics,and related topics. Ideal for self-teaching, this book is highly recommended forworking engineers, technical staff, students of physical optics andsignal analysis, and Fourier novices in alTable of ContentsPartial table of contents: Some of the How and Why of Fourier Analysis. Fourier Series and Spectra in One-Dimension for Functions of FinitePeriod. Fourier Series and Spectra for Functions of Infinite Period; One Dimensional. Fourier Spectra for Non-Periodic Functions; One-Dimensional. The Diffraction of Light and Fourier Transforms in TwoDimensions. A Brief Summary of Linear Systems Theory Applied to OpticalImaging. Fourier Optical Transformations by Computer. Apodization and Super-Resolution, Phase from Shift, and MultipleApertures. Complex Apertures. Operations in the Fourier Transform Plane. Other Interesting and Related Topics. References. A Selected Bibliography. Index.

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Book SynopsisA breakthrough approach to the theory and applications of stochastic integration The theory of stochastic integration has become an intensely studied topic in recent years, owing to its extraordinarily successful application to financial mathematics, stochastic differential equations, and more.Trade Review"...an important tool...gives the newest results in this field...shows an important application of vector integration..." (Bulletin of the Belgian Mathematical Society, Vol 11(1), 2004) "...it can be expected that...just like the author's 1967 volume, this book will stimulate further research on vector stochastic integration and can serve as a graduate-level reference work." (Mathematical Reviews Issue 2001h) "Dense, detailed, comprehensive introduction. Contains...material only found before in journals..." (American Mathematical Monthly, March 2002) "...a highly technical book." (The Mathematical Gazette, March 2002) "The author of this important and interesting book is a well-known specialist on vector measures." (Zentralblatt Math, Vol.974, No. 24 2001)Table of ContentsVector Integration. The Stochastic Integral. Martingales. Processes with Finite Variation. Processes with Finite Semivariation. The Itô Formula. Stochastic Integration in the Plane. Two-Parameter Martingales. Two-Parameter Processes with Finite Variation. Two-Parameter Processes with Finite Semivariation. References.

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Book SynopsisDevelops the higher parts of function theory in a unified presentation. Starts with elliptic integrals and functions and uniformization theory, continues with automorphic functions and the theory of abelian integrals and ends with the theory of abelian functions and modular functions in several variables. The last topic originates with the author and appears here for the first time in book form.Table of ContentsABELIAN FUNCTIONS. Power Series in Several Variables. The Preparation Theorem. Regular Functions. Meromorphic Functions. The Theorem of Weierstrass and Cousin. The Period Group. Jacobian Functions. Linearization of the Exponent System. The Period Relations. The Reduced Exponent System. Existence Proofs. Picard Varieties. The Addition Theorem. MODULAR FUNCTIONS OF SEVERAL VARIABLES. Automorphic Functions of Several Variables. Algebraic Relations Between Automorphic Functions. Symplectic Geometry. Abelian Functions and Modular Functions. The Fundamental Region of the Modular Group. Modular Forms. The Field of Modular Functions. Algebraic Dependence. Bibliography. Cumulative Index Volumes I, II, and III.

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Book SynopsisIncludes sections on the spectral resolution and spectral representation of self adjoint operators, invariant subspaces, strongly continuous one-parameter semigroups, the index of operators, the trace formula of Lidskii, the Fredholm determinant, and more.Trade Review"...an excellent source of facts for anyone working in functional analysis or operator theory." (Journal of Operator Theory, Vol.53, No.1, 2005) "For years Lax has been counted among the world's very top people in PDEs, so no serious student can afford to ignore his view of the foundations leading up to that subject." (Choice, Vol. 40, No. 4, December 2002) "...attractive...well suited for graduate courses...and useful for research mathematicians." (Mathematical Reviews, 2003a) "...The book is highly recommended to all students of analysis". (Zentralblatt MATH, Vol.1009, No.9, 2003) "A lot of good material, doled out in short chapters." (American Mathematical Monthly, August/September 2003)Table of ContentsForeword. Linear Spaces. Linear Maps. The Hahn-Banach Theorem. Applications of the Hahn-Banach Theorem. Normed Linear Spaces. Hilbert Space. Applications of Hilbert Space Results. Duals of Normed Linear Space. Applications of Duality. Weak Convergence. Applications of Weak Convergence. The Weak and Weak* Topologies. Locally Convex Topologies and the Krein-Milman Theorem. Examples of Convex Sets and their Extreme Points. Bounded Linear Maps. Examples of Bounded Linear Maps. Banach Algebras and their Elementary Spectral Theory. Gelfand's Theory of Commutative Banach Algebras. Applications of Gelfand's Theory of Commutative Banach Algebras. Examples of Operators and their Spectra. Compact Maps. Examples of Compact Operators. Positive Compact Operators. Fredholm's Theory of Integral Equations. Invariant Subspaces. Harmonic Analysis on a Halfline. Index Theory. Compact Symmetric Operators in Hilbert Space. Examples of Compact Symmetric Operators. Trace Class and Trace Formula. Spectral Theory of Symmetric, Normal and Unitary Operators. Spectral Theory of Self-Adjoint Operators. Examples of Self-Adjoint Operators. Semigroups of Operators. Groups of Unitary Operators. Examples of Strongly Continuous Semigroups. Scattering Theory. A Theorem of Beurling. Appendix A: The Riesz-Kakutani Representation Theorem. Appendix B: Theory of Distributions. Appendix C: Zorn's Lemma. Author Index. Subject Index.

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Book SynopsisFollowing on from the success of Fractal Geometry: Mathematical Foundations and Applications, this new sequel presents a variety of techniques in current use for studying the mathematics of fractals. Much of the material presented in this book has come to the fore in recent years.Table of ContentsMathematical Background. Review of Fractal Geometry. Some Techniques for Studying Dimension. Cookie-cutters and Bounded Distortion. The Thermodynamic Formalism. The Ergodic Theorem and Fractals. The Renewal Theorem and Fractals. Martingales and Fractals. Tangent Measures. Dimensions of Measures. Some Multifractal Analysis. Fractals and Differential Equations. References. Index.

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Book SynopsisThe discovery of the Fractional Fourier Transform and its role in optics provides an elegant mathematical framework within which to discuss diffraction and other fundamental aspects of optical systems. Easily-accessible, the reference work will serve as the standard reference on Fourier Transforms for many years to come.Trade Review"...[the authors] explain the basic concepts from various perspectives and survey its application in two areas where it is widely used." (SciTech Book News, Vol. 25, No. 4, December 2001)Table of ContentsPreface. Acknowledgments. Introduction. Signals, Systems, and Transformations. Wigner Distributions and Linear Canonical Transforms. The Fractional Fourier Transform. Time-Order and Space-Order Representations. The Discrete Fractional Fourier Transform. Optical Signals and Systems. Phase-Space Optics. The Fractional Fourier Transform in Optics. Applications of the Fractional Fourier Transform to Filtering, Estimation, and Signal Recovery. Applications of the Fractional Fourier Transform to Matched Filtering, Detection, and Pattern Recognition. Bibliography on the Fractional Fourier Transform. Other Cited Works. Credits. Index.

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Book SynopsisTable of ContentsPreface. Overview. The Discrete Fourier Transform. The Fast Fourier Transform. Weighting Functions. Frequency Analysis. Linear Filtering and Pattern Matching. Multidimentional Processing. Building-Block Algorithms. Algorithm Construction. Arithmetic Building Blocks for Architectures. Multiprocessor Architectures. Algorithm and Data Mappings. Arithmetic Formats. Chips. Board Decisions and Selection. Test. Design Examples. Glossary. Appendix: Table of Comparison Matrices. Index.

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Book SynopsisThis book is the first of a set dedicated to the mathematical tools used in partial differential equations derived from physics. Its focus is on normed or semi-normed vector spaces, including the spaces of Banach, Fréchet and Hilbert, with new developments on Neumann spaces, but also on extractable spaces. The author presents the main properties of these spaces, which are useful for the construction of Lebesgue and Sobolev distributions with real or vector values and for solving partial differential equations. Differential calculus is also extended to semi-normed spaces. Simple methods, semi-norms, sequential properties and others are discussed, making these tools accessible to the greatest number of students – doctoral students, postgraduate students – engineers and researchers without restricting or generalizing the results.Table of ContentsIntroduction xi Familiarization with Semi-normed Spaces xv Notations xvii Chapter 1 Prerequisites 1 1.1 Sets, mappings, orders 1 1.2 Countability 3 1.3 Construction of R 4 1.4 Properties of R 5 Part 1 Semi-normed Spaces 9 Chapter 2 Semi-normed Spaces 11 2.1 Definition of semi-normed spaces 11 2.2 Convergent sequences 15 2.3 Bounded, open and closed sets 17 2.4 Interior, closure, balls and semi-balls 21 2.5 Density, separability 23 2.6 Compact sets 25 2.7 Connected and convex sets 30 Chapter 3 Comparison of Semi-normed Spaces 33 3.1 Equivalent families of semi-norms 33 3.2 Topological equalities and inclusions 34 3.3 Topological subspaces 39 3.4 Filtering families of semi-norms 43 3.5 Sums of sets 46 Chapter 4 Banach, Fréchet and Neumann Spaces 49 4.1 Metrizable spaces 49 4.2 Properties of sets in metrizable spaces 51 4.3 Banach, Fréchet and Neumann spaces 55 4.4 Compacts sets in Fréchet spaces 57 4.5 Properties of R 58 4.6 Convergent sequences 60 4.7 Sequential completion of a semi-normed space 62 Chapter 5 Hilbert Spaces 65 5.1 Hilbert spaces 65 5.2 Projection in a Hilbert space 68 5.3 The space Rd 70 Chapter 6 Product, Intersection, Sum and Quotient of Spaces 73 6.1 Product of semi-normed spaces 73 6.2 Product of a semi-normed space by itself 78 6.3 Intersection of semi-normed spaces 80 6.4 Sum of semi-normed spaces 83 6.5 Direct sum of semi-normed spaces 89 6.6 Quotient space 93 Part 2 Continuous Mappings 95 Chapter 7 Continuous Mappings 97 7.1 Continuous mappings 97 7.2 Continuity and change of topology or restriction 100 7.3 Continuity of composite mappings 102 7.4 Continuous semi-norms 102 7.5 Continuous linear mappings 104 7.6 Continuous multilinear mappings 107 7.7 Some continuous mappings 111 Chapter 8 Images of Sets Under Continuous Mappings 115 8.1 Images of open and closed sets 115 8.2 Images of dense, separable and connected sets 117 8.3 Images of compact sets 119 8.4 Images under continuous linear mappings 121 8.5 Continuous mappings in compact sets 123 8.6 Continuous real mappings 124 8.7 Compacting mappings 125 Chapter 9 Properties of Mappings in Metrizable Spaces 129 9.1 Continuous mappings in metrizable spaces 129 9.2 Banach’s fixed point theorem 133 9.3 Baire’s theorem 134 9.4 Open mapping theorem 136 9.5 Banach–Schauder’s continuity theorem 138 9.6 Closed graph theorem 139 Chapter 10 Extension of Mappings, Equicontinuity 141 10.1 Extension of equalities by continuity 141 10.2 Continuous extension of mappings 142 10.3 Equicontinuous families of mappings 146 10.4 Banach–Steinhaus equicontinuity theorem 148 Chapter 11 Compactness in Mapping Spaces 153 11.1 The spaces F(X; F) and C(X; F)-pt 153 11.2 Zorn’s lemma 154 11.3 Compactness in F(X; F) 157 11.4 An Ascoli compactness theorem in C(X; F)-pt 161 Chapter 12 Spaces of Linear or Multilinear Mappings 163 12.1 The space L(E; F) 163 12.2 Bounded sets in L(E; F) 165 12.3 Sequential completeness of L(E; F) when E is metrizable 167 12.4 Semi-norms and norm on L(E; F) when E isnormed 169 12.5 Continuity of the composition of linear mappings 171 12.6 Inversibility in the neighborhood of an isomorphism 174 12.7 The space Ld(E1 × ··· × Ed; F) 178 12.8 Separation of the variables of a multilinear mapping 181 Part 3 Weak Topologies 187 Chapter 13 Duality 189 13.1 Dual 189 13.2 Dual of a metrizable or normed space 193 13.3 Dual of a Hilbert space 196 13.4 Extraction of ∗ weakly converging subsequences 199 13.5 Continuity of the bilinear form of duality 203 13.6 Dual of a product 205 13.7 Dual of a direct sum 206 Chapter 14 Dual of a Subspace 209 14.1 Hahn–Banach theorem 209 14.2 Corollaries of the Hahn–Banach theorem 211 14.3 Characterization of a dense subspace 212 14.4 Dual of a subspace 213 14.5 Dual of an intersection 215 14.6 Dangerous identifications 216 Chapter 15 Weak Topology 221 15.1 Weak topology 221 15.2 Weak continuity and topological inclusions 224 15.3 Weak topology of a product 225 15.4 Weak topology of an intersection 226 15.5 Norm and semi-norms of a weak limit 228 Chapter 16 Properties of Sets for the Weak Topology 231 16.1 Banach–Mackey theorem (weakly bounded sets) 231 16.2 Gauge of a convex open set 233 16.3 Mazur’s theorem (weakly closed convex sets) 235 16.4 ˘Smulian’s theorem (weakly compact sets) 237 16.5 Semi-weak continuity of a bilinear mapping 240 Chapter 17 Reflexivity 243 17.1 Reflexive spaces 243 17.2 Sequential completion of a semi-reflexive space 247 17.3 Prereflexivity of metrizable spaces 248 17.4 Reflexivity of Hilbert spaces 250 17.5 Reflexivity of uniformly convex Banach spaces 252 17.6 A property of the combinations of linear forms 256 17.7 Characterizations of semi-reflexivity 257 17.8 Reflexivity of a subspace 261 17.9 Reflexivity of the image of a space 261 17.10 Reflexivity of the dual 263 Chapter 18 Extractable Spaces 265 18.1 Extractable spaces 265 18.2 Extractability of Hilbert spaces 266 18.3 Extractability of semi-reflexive spaces 267 18.4 Extractability of a subspace or of the image of a space 269 18.5 Extractability of a product or of a sum of spaces 270 18.6 Extractability of an intersection of spaces 271 18.7 Sequential completion of extractable spaces 271 Part 4 Differential Calculus 273 Chapter 19 Differentiable Mappings 275 19.1 Differentiable mappings 275 19.2 Differentiality, continuity and linearity 277 19.3 Differentiation and change of topology or restriction 279 19.4 Mean value theorem 281 19.5 Bounds on a real differentiable mapping 284 19.6 Differentiation of a composite mapping 286 19.7 Differential of an inverse mapping 289 19.8 Inverse mapping theorem 290 Chapter 20 Differentiation of Multivariable Mappings 295 20.1 Partial differentiation 295 20.2 Differentiation of a multilinear or multi-component mapping 298 20.3 Differentiation of a composite multilinear mapping 300 Chapter 21 Successive Differentiations 303 21.1 Successive differentiations 303 21.2 Schwarz’s symmetry principle 305 21.3 Successive differentiations of a composite mapping 308 Chapter 22 Derivation of Functions of One Real Variable 313 22.1 Derivative of a function of one real variable 313 22.2 Derivative of a real function of one real variable 315 22.3 Leibniz formula 319 22.4 Derivatives of the power, logarithm and exponential functions 320 Bibliography 325 Cited Authors 331 Index 335

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Book SynopsisIn recent years there has been a surge of profound new developments in various aspects of analysis whose connecting thread is the use of Banach space methods. Indeed, many problems seemingly far from the classical geometry of Banach spaces have been solved using Banach space techniques. This volume contains papers by participants of the conference "Banach Spaces and their Applications in Analysis", held in May 2006 at Miami University in Oxford, Ohio, in honor of Nigel Kalton's 60th birthday. In addition to research articles contributed by participants, the volume includes invited expository articles by principal speakers of the conference, who are leaders in their areas. These articles present overviews of new developments in each of the conference's main areas of emphasis, namely nonlinear theory, isomorphic theory of Banach spaces including connections with combinatorics and set theory, algebraic and homological methods in Banach spaces, approximation theory and algorithms in Banach spaces. This volume also contains an expository article about the deep and broad mathematical work of Nigel Kalton, written by his long time collaborator, Gilles Godefroy. Godefroy's article, and in fact the entire volume, illustrates the power and versatility of applications of Banach space methods and underlying connections between seemingly distant areas of analysis.

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Book SynopsisThis is the first part of the second revised and extended edition of the well established book "Function Spaces" by Alois Kufner, Oldřich John, and Svatopluk Fučík. Like the first edition this monograph is an introduction to function spaces defined in terms of differentiability and integrability classes. It provides a catalogue of various spaces and benefits as a handbook for those who use function spaces in their research or lecture courses. This first volume is devoted to the study of function spaces, based on intrinsic properties of a function such as its size, continuity, smoothness, various forms of a control over the mean oscillation, and so on. The second volume will be dedicated to the study of function spaces of Sobolev type, in which the key notion is the weak derivative of a function of several variables.

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Book SynopsisAs in the field of "Invariant Distances and Metrics in Complex Analysis" there was and is a continuous progress this is now the second extended edition of the corresponding monograph. This comprehensive book is about the study of invariant pseudodistances (non-negative functions on pairs of points) and pseudometrics (non-negative functions on the tangent bundle) in several complex variables. It is an overview over a highly active research area at the borderline between complex analysis, functional analysis and differential geometry. New chapters are covering the Wu, Bergman and several other metrics. The book considers only domains in Cn and assumes a basic knowledge of several complex variables. It is a valuable reference work for the expert but is also accessible to readers who are knowledgeable about several complex variables. Each chapter starts with a brief summary of its contents and continues with a short introduction. It ends with an "Exercises" and a "List of problems" section that gathers all the problems from the chapter. The authors have been highly successful in giving a rigorous but readable account of the main lines of development in this area.

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Book SynopsisRegularization methods aimed at finding stable approximate solutions are a necessary tool to tackle inverse and ill-posed problems. Inverse problems arise in a large variety of applications ranging from medical imaging and non-destructive testing via finance to systems biology. Many of these problems belong to the class of parameter identification problems in partial differential equations (PDEs) and thus are computationally demanding and mathematically challenging. Hence there is a substantial need for stable and efficient solvers for this kind of problems as well as for a rigorous convergence analysis of these methods. This monograph consists of five parts. Part I motivates the importance of developing and analyzing regularization methods in Banach spaces by presenting four applications which intrinsically demand for a Banach space setting and giving a brief glimpse of sparsity constraints. Part II summarizes all mathematical tools that are necessary to carry out an analysis in Banach spaces. Part III represents the current state-of-the-art concerning Tikhonov regularization in Banach spaces. Part IV about iterative regularization methods is concerned with linear operator equations and the iterative solution of nonlinear operator equations by gradient type methods and the iteratively regularized Gauß-Newton method. Part V finally outlines the method of approximate inverse which is based on the efficient evaluation of the measured data with reconstruction kernels.

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Book SynopsisMost classes of operators that are not isomorphic embeddings are characterized by some kind of a “smallness” condition. Narrow operators are those operators defined on function spaces that are “small” at {-1,0,1}-valued functions, e.g. compact operators are narrow. The original motivation to consider such operators came from theory of embeddings of Banach spaces, but since then they were also applied to the study of the Daugavet property and to other geometrical problems of functional analysis. The question of when a sum of two narrow operators is narrow, has led to deep developments of the theory of narrow operators, including an extension of the notion to vector lattices and investigations of connections to regular operators. Narrow operators were a subject of numerous investigations during the last 30 years. This monograph provides a comprehensive presentation putting them in context of modern theory. It gives an in depth systematic exposition of concepts related to and influenced by narrow operators, starting from basic results and building up to most recent developments. The authors include a complete bibliography and many attractive open problems.Table of ContentsChapter 1. Preliminaries Chapter 2. Each small operator is narrow Chapter 3. Applications to the geometry of Lp spaces for 0 < p < 1 5 Chapter 4. A very non-compact narrow operator Chapter 5. Some deep results on narrow operators Chapter 6. Weak embeddings of L1 Chapter 7. For what spaces X every operator T 2 L(Lp;X) is narrow? Chapter 8. Ideal properties of narrow operators Chapter 9. Daugavet type properties of Lorentz spaces with Chapter 10. Narrow operators on vector lattices Chapter 11. Some generalizations of narrow operators Bibliography

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Book SynopsisThis monograph contains a study on various function classes, a number of new results and new or easy proofs of old results (Fefferman-Stein theorem on subharmonic behavior, theorems on conjugate functions and fractional integration on Bergman spaces, Fefferman's duality theorem), which are interesting for specialists; applications of the Hardy-Littlewood inequalities on Taylor coefficients to (C, α)-maximal theorems and (C, α)-convergence; a study of BMOA, due to Knese, based only on Green's formula; the problem of membership of singular inner functions in Besov and Hardy-Sobolev spaces; a full discussion of g-function (all p > 0) and Calderón's area theorem; a new proof, due to Astala and Koskela, of the Littlewood-Paley inequality for univalent functions; and new results and proofs on Lipschitz spaces, coefficient multipliers and duality, including compact multipliers and multipliers on spaces with non-normal weights. It also contains a discussion of analytic functions and lacunary series with values in quasi-Banach spaces with applications to function spaces and composition operators. Sixteen open questions are posed. The reader is assumed to have a good foundation in Lebesgue integration, complex analysis, functional analysis, and Fourier series. Further information can be found at the author's website at http://poincare.matf.bg.ac.rs/~pavlovic.Table of ContentsPreface 1 Quasi-Banach spaces 1.1 Quasinorm and p-norm 1.2 Linear operators 1.3 The closed graph theorem The open mapping theorem The uniform boundedness principle The closed graph theorem 1.4 F-spaces 1.5 The spaces lp 1.6 Spaces of analytic functions 1.7 The Abel dual of a space of analytic functions 1.7a Homogeneous spaces 2 Interpolation and maximal functions 2.1 The Riesz/Thorin theorem 2.2 Weak Lp-spaces and Marcinkiewicz’s theorem 2.3 The maximal function and Lebesgue points 2.4 The Rademacher functions and Khintchine’s inequality 2.5 Nikishin’s theorem 2.6 Nikishin and Stein’s theorem 2.7 Banach’s principle, the theorem on a.e. convergence, and Sawier’s theorems 2.8 Addendum: Vector-valued maximal theorem 3 Poisson integral 3.1 Harmonic functions 3.1a Green’s formulas 3.1b The Poisson integral 3.2 Borel measures and the space h1 3.3 Positive harmonic functions 3.4 Radial and non-tangential limits of the Poisson integral 3.4a Convolution of harmonic functions 3.5 The spaces hp and Lp(T) 3.6 A theorem of Littlewood and Paley 3.7 Harmonic Schwarz lemma 4 Subharmonic functions 4.1 Basic properties 4.1a The maximum principle 4.1b Approximation by smooth functions 4.2 Properties of the mean values 4.3 Integral means of univalent functions Prawitz’ theorem Distortion theorems 4.4 The subordination principle 4.5 The Riesz measure Green’s formula The Riesz measure of | f |p (f є H(D)) and | u |p (u є hp) 5 Classical Hardy spaces 5.1 Basic properties The decomposition lemma of Hardy and Littlewood 5.1a Radial limits The Poisson integral of log | f* | 5.2 The space H1 5.3 Blaschke products Riesz’ factorization theorem 5.4 Some inequalities 5.5 Inner and outer functions 5.5a Beurling’s approximation theorem 5.6 Composition with inner functions. Stephenson’s theorems 5.6a Approximation by inner functions 6 Conjugate functions 6.1 Harmonic conjugates 6.1a The Privalov/Plessner theorem and the Hilbert operator 6.2 Riesz projection theorem 6.2a The Hardy/Stein identity 6.2b Proof of Riesz’ theorems 6.3 Applications of the projection theorem 6.4 Aleksandrov’s theorem: Lp(T) = Hp(T) + \overline{Hp}(T) 6.5 Strong convergence in H1 6.6 Quasiconformal harmonic homeomorphisms and the Hilbert transformation 7 Maximal functions, interpolation, and coefficients 7.1 Maximal theorems 7.1a Hardy/Littlewood/Sobolev theorem 7.2 Maximal characterization of Hp (Burkholder, Gundy and Silverstein) 7.3 “Smooth” Cesàro means σα-maximal theorem The “W-maximal” theorem 7.4 Interpolation of operators on Hardy spaces 7.4a Application to Taylor coefficients and mean growth 7.4b On the Hardy/Littlewood inequality 7.4c The case of monotone coefficients 7.5 Lacunary series 7.6 A proof of the σα-maximal theorem 8 Bergman spaces: Atomic decomposition 8.1 Bergman spaces 8.2 Reproducing kernels 8.3 The Coifman/Rochberg theorem q-envelops of Hardy spaces 8.4 Coefficients of vector-valued functions. Kalton’s theorems 8.4a Inequalities for a Hadamard product 8.4b Applications to spaces of scalar valued functions 9 Lipschitz spaces 9.1 Lipschitz spaces of first order 9.2 Conjugate functions 9.3 Lipschitz condition for the modulus. Dyakonov’s theorems with simple proofs by Pavlovic 9.4 Lipschitz spaces of higher order 9.5 Lipschitz spaces as duals of Hp, p < 1 10 Generalized Bergman spaces and Besov spaces 10.1 Decomposition of mixed norm spaces: case 1 < p < ∞ 10.1a Besov spaces 10.2 Decomposition of mixed norm spaces: case 0 < p ≤ ∞ 10.2a Radial limits of Hardy/Bloch functions 10.2b Fractional integration and differentiation 10.3 Möbius invariant Besov spaces 10.4 Mean Lipschitz spaces 10.4a Lacunary series in mixed norm spaces 10.5 Duality in the case 0 < p ≤ ∞ 10.6 Appendix: Characterizations of Besov spaces 11 BMOA, Bloch space 11.1 The dual of H1 and the Carleson measures Proof of Fefferman’s theorem 11.2 Vanishing mean osillation 11.3 BMOA and mean Lipschitz spaces 11.4 Coefficients of BMOA-functions 11.4a Lacunary series 11.5 The Bloch space 11.5a On the predual of B Functions with decreasing coefficients 12 Subharmonic behavior 12.1 Subharmonic behavior and Bergman spaces Two simple proofs of Hardy/Littlewood/Fefferman/Stein theorem 12.2 The space hp, p < 1 Two open problems posed by Hardy and Littlewood 12.3 Subharmonic behavior of smooth functions 12.3a Quasi-nearly subharmonic functions 12.3b Regularly oscillating functions 12.4 A generalization of the Littlewood/Paley theorem 12.4a Invariant Besov spaces and the derivatives of the integral means 12.4b Addendum: The case of vector valued functions 12.5 Mixed norm spaces of harmonic functions 13 Littlewood/Paley theory 13.1 Some more vector maximal functions 13.2 The Littlewood/Paley g-function Calderon’s generalization of the area theorem (p > 0) A proof of a the Littlewood/Paley g-theorem (p > 0) 13.3 Applications of Cesàro means 13.4 The Littlewood/Paley g-theorem in a generalized form An improvement 13.5 Proof of Calderon’s theorem 14 Tauberian theorems and lacunary series on the interval (0,1) 14.1 Karamata’s theorem and Littlewood’s theorem 14.1a Tauberian nature of Λp1/p 14.2 Lacunary series in C[0,1] 14.2a Lacunary series on weighted L∞-spaces 14.3 Lp-integrability of lacunary series on (0,1) 14.3a Some consequences Bibliography

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Book SynopsisIn the last 40 years semi-linear elliptic equations became a central subject of study in the theory of nonlinear partial differential equations. On the one hand, the interest in this area is of a theoretical nature, due to its deep relations to other branches of mathematics, especially linear and nonlinear harmonic analysis, dynamical systems, differential geometry and probability. On the other hand, this study is of interest because of its applications. Equations of this type come up in various areas such as problems of physics and astrophysics, curvature problems in Riemannian geometry, logistic problems related for instance to population models and, most importantly, the study of branching processes and superdiffusions in the theory of probability. The aim of this book is to present a comprehensive study of boundary value problems for linear and semi-linear second order elliptic equations with measure data. We are particularly interested in semi-linear equations with absorption. The interactions between the diffusion operator and the absorption term give rise to a large class of nonlinear phenomena in the study of which singularities and boundary trace play a central role. This book is accessible to graduate students and researchers with a background in real analysis and partial differential equations.Table of Contents1 Linear second order elliptic equations with measure data 5 1.1 Linear boundary value problems with L1 data. . . . . . . . . . . . . 5 1.2 Measure data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 M-boundary trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.4 The Herglotz – Doob theorem . . . . . . . . . . . . . . . . . . . . . . 24 1.5 Sub-solutions, super-solutions and Kato’s inequality. . . . . . . . . . 26 1.6 Boundary Harnack principle. . . . . . . . . . . . . . . . . . . . . . . 36 1.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2 Nonlinear second order elliptic equations with measure data 43 2.1 Semilinear problems with L1 data . . . . . . . . . . . . . . . . . . . . 43 2.2 Semilinear problems with bounded measure data . . . . . . . . . . . 47 2.3 Subcritical non-linearities . . . . . . . . . . . . . . . . . . . . . . . . 55 2.3.1 Weak Lp spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.3.2 Continuity of G and P relative to Lp w norm . . . . . . . . . . 59 2.3.3 Continuity of a superposition operator. . . . . . . . . . . . . 61 2.3.4 Weak continuity of Sg . . . . . . . . . . . . . . . . . . . . . . . 65 2.3.5 Weak continuity of Sg @ . . . . . . . . . . . . . . . . . . . . . . 69 2.4 The structure of Mg. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 2.5 Remarks on unbounded domains . . . . . . . . . . . . . . . . . . . . 80 2.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3 The boundary trace and associated boundary value problems. 83 3.1 The boundary trace . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.1.1 Moderate solutions . . . . . . . . . . . . . . . . . . . . . . . . 83 3.1.2 Positive solutions . . . . . . . . . . . . . . . . . . . . . . . . . 883.1.3 Unbounded domains . . . . . . . . . . . . . . . . . . . . . . . 98 3.2 Maximal solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.3 The boundary value problem with rough trace. . . . . . . . . . . . . 101 3.4 A problem with fading absorption. . . . . . . . . . . . . . . . . . . . 108 3.4.1 The similarity transformation and an extension of the Keller – Osserman estimate. . . . . . . . . . . . . . . . . . . . . . . 109 3.4.2 Barriers and maximal solutions. . . . . . . . . . . . . . . . . . 111 3.4.3 The critical exponent. . . . . . . . . . . . . . . . . . . . . . . 116 3.4.4 The very singular solution. . . . . . . . . . . . . . . . . . . . 119 3.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4 Isolated singularities 133 4.1 Universal upper bounds . . . . . . . . . . . . . . . . . . . . . . . . . 133 4.1.1 The Keller-Osserman estimates . . . . . . . . . . . . . . . . . 133 4.1.2 Applications to model cases . . . . . . . . . . . . . . . . . . 138 4.2 Isolated singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 4.2.1 Removable singularities . . . . . . . . . . . . . . . . . . . . . 140 4.2.2 Isolated positive singularities . . . . . . . . . . . . . . . . . . 142 4.2.3 Isolated signed singularities . . . . . . . . . . . . . . . . . . . 151 4.3 Boundary singularities . . . . . . . . . . . . . . . . . . . . . . . . . . 158 4.3.1 Upper bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 4.3.2 The half space case . . . . . . . . . . . . . . . . . . . . . . . . 160 4.3.3 The case of a general domain . . . . . . . . . . . . . . . . . . 167 4.4 Boundary singularities with fading absorption . . . . . . . . . . . . . 176 4.4.1 Power-type degeneracy . . . . . . . . . . . . . . . . . . . . . . 176 4.4.2 A strongly fading absorption . . . . . . . . . . . . . . . . . . 180 4.5 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 4.5.1 General results of isotropy . . . . . . . . . . . . . . . . . . . . 187 4.5.2 Isolated singularities of super-solutions . . . . . . . . . . . . 188 4.6 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 5 Classical theory of maximal and large solutions 195 5.1 Maximal solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 5.1.1 Global conditions . . . . . . . . . . . . . . . . . . . . . . . . . 195 5.1.2 Local conditions . . . . . . . . . . . . . . . . . . . . . . . . . 200 5.2 Large solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 5.2.1 General nonlinearities . . . . . . . . . . . . . . . . . . . . . . 2015.2.2 The power and exponential cases . . . . . . . . . . . . . . . . 206 5.3 Uniqueness of large solutions . . . . . . . . . . . . . . . . . . . . . . 210 5.3.1 General uniqueness results . . . . . . . . . . . . . . . . . . . . 211 5.3.2 Applications to power and exponential types nonlinearities . 219 5.4 Equations with forcing term . . . . . . . . . . . . . . . . . . . . . . . 221 5.4.1 Maximal and minimal large solutions . . . . . . . . . . . . . . 222 5.4.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 5.5 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 6 Further results on singularities and large solutions 233 6.1 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 6.1.1 Internal singularities . . . . . . . . . . . . . . . . . . . . . . . 233 6.1.2 Boundary singularities . . . . . . . . . . . . . . . . . . . . . . 244 6.2 Symmetries of large solutions . . . . . . . . . . . . . . . . . . . . . . 259 6.3 Sharp blow-up rate of large solutions . . . . . . . . . . . . . . . . . . 268 6.3.1 Estimates in an annulus . . . . . . . . . . . . . . . . . . . . . 269 6.3.2 Curvature secondary effects . . . . . . . . . . . . . . . . . . . 275 6.4 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

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Book Synopsis Dieses vierfarbige Lehrbuch wendet sich an Studierende der Mathematik in Bachelor-Studiengängen. Es bietet in einem Band ein lebendiges Bild der mathematischen Inhalte, die üblicherweise im zweiten und dritten Studienjahr behandelt werden (mit Ausnahme der Algebra).Mathematik-Studierende finden wichtige Begriffe, Sätze und Beweise ausführlich und mit vielen Beispielen erklärt und werden an grundlegende Konzepte und Methoden herangeführt.Im Mittelpunkt stehen das Verständnis der mathematischen Zusammenhänge und des Aufbaus der Theorie sowie die Strukturen und Ideen wichtiger Sätze und Beweise. Es wird nicht nur ein in sich geschlossenes Theoriengebäude dargestellt, sondern auch verdeutlicht, wie es entsteht und wozu die Inhalte später benötigt werden.Herausragende Merkmale sind: durchgängig vierfarbiges Layout mit mehr als 350 Abbildungen prägnant formulierte Kerngedanken bilden die Abschnittsüberschriften Selbsttests in kurzen Abständen ermöglichen Lernkontrollen während des Lesens farbige Merkkästen heben das Wichtigste hervor „Unter-der-Lupe“-Boxen zoomen in Beweise hinein, motivieren und erklären Details „Hintergrund-und-Ausblick“-Boxen stellen Zusammenhänge zu anderen Gebieten und weiterführenden Themen her Zusammenfassungen zu jedem Kapitel sowie Übersichtsboxen mehr als 500 Verständnisfragen, Rechenaufgaben und Aufgaben zu Beweisen Der inhaltliche Schwerpunkt liegt auf dem weiteren Ausbau der Analysis sowie auf den Themen der Vorlesungen Numerik sowie Wahrscheinlichkeitstheorie und Statistik. Behandelt werden darüber hinaus Inhalte und Methodenkompetenzen, die vielerorts im zweiten und dritten Studienjahr der Mathematikausbildung vermittelt werden.Auf der Website zum Buch Matheweb finden Sie Hinweise, Lösungswege und Ergebnisse zu allen Aufgaben die Möglichkeit, zu den Kapiteln Fragen zu stellen Das Buch wird allen Studierenden der Mathematik ein verlässlicher Begleiter sein.Trade Review“ ... Ausführliche Erklärungen und über 400 Abbildungen verdeutlichen abstrakte Sachverhalte, kompakte Übersichten liefern zentrale Ergebnisse, Kontrollfragen ermöglichen eine fortlaufende Verständniskontrolle und Übungsaufgaben dienen der eingehenden Beschäftigung mit dem Stoff ... Der Zielgruppe als Lehrbuch und Nachschlagewerk auch neben der Studienliteratur zu den einzelnen Teilgebieten sehr dienlich.” (Philipp Kastendieck, in: ekz-Informationsdienst, Jg. 11, 2016)“... für den Autodidakt kann dieses Lehrbuch empfohlen werden, da eine Vielfalt an Beispielen , Übungsaufgaben und entsprechenden Abfragen den Einstieg im nicht immer leichten Lehrstoff erleichtert. ... kann dieses Lehrbuch für das Mathematikstudium empfohlen werden. Es beinhaltet die Grundlagen des Mathematikstudiums und hat den Vorteil ...” (La, in: Amazon.de, 10. November 2015)Table of Contents1 Mathematik – eine Wissenschaft für sich.- 2 Lineare Differenzialgleichungen – Systeme und Gleichungen höherer Ordnung.- 3 Randwertprobleme und nichtlineare Differenzialgleichungen – Funktionen sind gesucht.- 4 Qualitative Theorie – jenseits von analytischen und mehr als numerische Lösungen.- 5 Funktionentheorie – Analysis im Komplexen.- 6 Differenzialformen und der allgemeine Satz von Stokes.- 7 Grundzüge der Maß- und Integrationstheorie vom Messen und Mitteln.- 8 Lineare Funktionalanalysis – Operatoren statt Matrizen.- 9 Fredholm-Gleichungen – kompakte Störungen der Identität.- 10 Hilberträume – fast wie im Anschauungsraum.- 11 Warum Numerische Mathematik? – Modellierung, Simulation und Optimierung.- 12 Interpolation – Splines und mehr.- 13 Quadratur – numerische Integrationsmethoden.- 14 Numerik linearer Gleichungssysteme – Millionen von Variablen im Griff.- 15 Eigenwertprobleme – Einschließen und Approximieren.- 16 Lineare Ausgleichsprobleme – im Mittel das Beste.- 17 Nichtlineare Gleichungen und Systeme – numerisch gelöst.- 18 Numerik gewöhnlicher Differenzialgleichungen – Schritt für Schritt zur Trajektorie.- 19 Wahrscheinlichkeitsräume – Modelle für stochastische Vorgänge.- 20 Bedingte Wahrscheinlichkeit und Unabhängigkeit – Meister Zufall hängt (oft) ab.- 21 Diskrete Verteilungsmodelle – wenn der Zufall zählt.- 22 Stetige Verteilungen und allgemeine Betrachtungen – jetzt wird es analytisch.- 23 Konvergenzbegriffe und Grenzwertsätze – Stochastik für große Stichproben.- 24 Grundlagen der Mathematischen Statistik – vom Schätzen und Testen.

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