Functional analysis and transforms Books

Book SynopsisMeasure and integration, metric spaces, the elements of functional analysis in Banach spaces, and spectral theory in Hilbert spaces â all in a single study. Detailed analyses. Problems. Bibliography. Index.

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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 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 SynopsisThis book provides the first lucid and accessible account to the modern study of the geometry of four-manifolds. It has become required reading for postgraduates and research workers whose research touches on this topic. Pre-requisites are a firm grounding in differential topology, and geometry as may be gained from the first year of a graduate course. The subject matter of this book is the most significant breakthrough in mathematics of the last fifty years, and Professor Donaldson won a Fields medal for his work in the area. The authors start from the standpoint that the fundamental group and intersection form of a four-manifold provides information about its homology and characteristic classes, but little of its differential topology. It turns out that the classification up to diffeomorphism of four-manifolds is very different from the classification of unimodular forms and that the study of this question leads naturally to the new Donaldson invariants of four-manifolds. A central tTrade Review... authoritative and comprehensive... it must be regarded as compulsory reading for any young researcher approaching this difficult but fascinating area. * Bulletin of the London Mathematical Society *Table of Contents1. Four-manifolds ; 2. Connections ; 3. The Fourier transform and ADHM construction ; 4. Yang-Mills moduli spaces ; 5. Topology and connections ; 6. Stable holomorphic bundles over Kahler surfaces ; 7. Excision and glueing ; 8. Non-existence results ; 9. Invariants of smooth four-manifolds ; 10. The differential topology of algebraic surfaces ; Appendix ; References ; Index

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Book SynopsisThis monograph treats the analysis of symmetric cones in a systematic way. It discusses harmonic analysis and special functions associated with symmetric cones; it also tries these results together with the study of holomorphic functions on bounded symmetric domains of tube type.Trade Review... the present book is more carefully directed at the graduate student level, includes numerous exercises, and has its emphasis more on the harmonic analysis side. Such a presentation is much needed. The detailed exposition, careful choice of organization and notation, and very helpful collection of exercises, mostly of medium difficulty, all attest to the effort put into this joint venture. As a highly readable and accessible presentation of Jordan algebras and their applications to Riemannian geometry and harmonic analysis, the book is strongly recommended to all analysts (starting at graduate level) working in the multi-variable setting of symmetric spaces and Lie groups. Bulletin of the London Mathematical SocietyTable of ContentsI. Convex cones ; II. Jordan algebras ; III. Symmetric cones and Euclidean Jordan algebras ; IV. The Peirce decomposition in a Jordan algebra ; V. Classification of Euclidean Jordan algebras ; VI. Polar decomposition and Gauss decomposition ; VII. The gamma function of a symmetric cone ; VIII. Complex Jordan algebras ; IX. Tube domains over convex cones ; X. Symmetric domains ; XI. Conical and spherical polynomials ; XII. Taylor and Laurent series ; XIII. Functions spaces on symmetric domains ; XIV. Invariant differential operators and spherical functions ; XV. Special functions ; XVI. Representations of Jordan algebras and Euclidean Fourier analysis ; Bibliography

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Book SynopsisThis monograph explores the equivalence of signal functions with their sets of values taken at discrete points. Beginning with an introduction to the main ideas, and background material on Fourier analysis and Hilbert spaces and their bases, it covers a wide variety of topics.Trade Review...the text is written by use of LATEX and its beautiful graphics reveal the power and the advantages of this system. * Zentralblatt fuer Mathematik 827/97 *Table of Contents1. An introduction to sampling theory ; 1.1 General introduction ; 1.2 Introduction - continued ; 1.3 The seventeenth to the mid twentieth century - a brief review ; 1.4 Interpolation and sampling from the seventeenth century to the mid twentieth century - a brief review ; 1.5 Introduction - concluding remarks ; 2. Background in Fourier analysis ; 2.1 The Fourier Series ; 2.2 The Fourier transform ; 2.3 Poisson's summation formula ; 2.4 Tempered distributions - some basic facts ; 3. Hilbert spaces, bases and frames ; 3.1 Bases for Banach and Hilbert spaces ; 3.2 Riesz bases and unconditional bases ; 3.3 Frames ; 3.4 Reproducing kernel Hilbert spaces ; 3.5 Direct sums of Hilbert spaces ; 3.6 Sampling and reproducing kernels ; 4. Finite sampling ; 4.1 A general setting for finite sampling ; 4.2 Sampling on the sphere ; 5. From finite to infinite sampling series ; 5.1 The change to infinite sampling series ; 5.2 The Theorem of Hinsen and Kloosters ; 6. Bernstein and Paley-Weiner spaces ; 6.1 Convolution and the cardinal series ; 6.2 Sampling and entire functions of polynomial growth ; 6.3 Paley-Weiner spaces ; 6.4 The cardinal series for Paley-Weiner spaces ; 6.5 The space ReH1 ; 6.6 The ordinary Paley-Weiner space and its reproducing kernel ; 6.7 A convergence principle for general Paley-Weiner spaces ; 7. More about Paley-Weiner spaces ; 7.1 Paley-Weiner theorems - a review ; 7.2 Bases for Paley-Weiner spaces ; 7.3 Operators on the Paley-Weiner space ; 7.4 Oscillatory properties of Paley-Weiner functions ; 8. Kramer's lemma ; 8.1 Kramer's Lemma ; 8.2 The Walsh sampling therem ; 9. Contour integral methods ; 9.1 The Paley-Weiner theorem ; 9.2 Some formulae of analysis and their equivalence ; 9.3 A general sampling theorem ; 10. Ireggular sampling ; 10.1 Sets of stable sampling, of interpolation and of uniqueness ; 10.2 Irregular sampling at minimal rate ; 10.3 Frames and over-sampling ; 11. Errors and aliasing ; 11.1 Errors ; 11.2 The time jitter error ; 11.3 The aliasing error ; 12. Multi-channel sampling ; 12.1 Single channel sampling ; 12.3 Two channels ; 13. Multi-band sampling ; 13.1 Regular sampling ; 13.3 An algorithm for the optimal regular sampling rate ; 13.4 Selectively tiled band regions ; 13.5 Harmonic signals ; 13.6 Band-ass sampling ; 14. Multi-dimensional sampling ; 14.1 Remarks on multi-dimensional Fourier analysis ; 14.2 The rectangular case ; 14.3 Regular multi-dimensional sampling ; 15. Sampling and eigenvalue problems ; 15.1 Preliminary facts ; 15.2 Direct and inverse Sturm-Liouville problems ; 15.3 Further types of eigenvalue problem - some examples ; 16. Campbell's generalised sampling theorem ; 16.1 L.L. Campbell's generalisation of the sampling theorem ; 16.2 Band-limited functions ; 16.3 Non band-limited functions - an example ; 17. Modelling, uncertainty and stable sampling ; 17.1 Remarks on signal modelling ; 17.2 Energy concentration ; 17.3 Prolate Spheroidal Wave functions ; 17.4 The uncertainty principle of signal theory ; 17.5 The Nyquist-Landau minimal sampling rate

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Book SynopsisSequence spaces and summability over valued fields is a research book aimed at research scholars, graduate students and teachers with an interest in Summability Theory both Classical (Archimedean) and Ultrametric (non-Archimedean).The book presents theory and methods in the chosen topic, spread over 8 chapters that seem to be important at research level in a still developing topic.Key Features Presented in a self-contained manner Provides examples and counterexamples in the relevant contexts Provides extensive references at the end of each chapter to enable the reader to do further research in the topic Presented in the same book, a comparative study of Archimedean and non-Archimedean Summability Theory Appeals to young researchers and experienced mathematicians who wish to explore new areas in Summability Theory The book is written by a very experienced educator and researcher in MatTable of ContentsAbout the Author. Foreword. Preface. Preliminaries. On Certain Spaces Containing the Space of Cauchy Sequences. Matrix Transformations Between Some Other Sequence Spaces. Characterization of Regular and Schur Matrices. A Study of the Sequence Space c0(p). On the Sequence Spaces `(p), c0(p), c(p), `1(p) over Non-archimedean Fields. A Characterization of the Matrix Class (`1; c0) and Summability Matrices of Type M in Non-archimedean Analysis. More Steinhaus Type Theorems over Valued Fields. Index.

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Book SynopsisThis concise text is intended as an introductory course in measure and integration. It covers essentials of the subject, providing ample motivation for new concepts and theorems in the form of discussion and remarks, and with many worked-out examples. The novelty of  Measure and Integration: A First Course is in its style of exposition of the standard material in a student-friendly manner. New concepts are introduced progressively from less abstract to more abstract so that the subject is felt on solid footing. The book starts with a review of Riemann integration as a motivation for the necessity of introducing the concepts of measure and integration in a general setting. Then the text slowly evolves from the concept of an outer measure of subsets of the set of real line to the concept of Lebesgue measurable sets and Lebesgue measure, and then to the concept of a measure, measurable function, and integration in a more general setting. Again, intTable of ContentsPreface. Note to the Reader. Review of Riemann Integral. Lebesgue Measure. Measure and Measurable Functions. Integral of Positive Measurable Functions. Integral of Complex Measurable Functions. Integration on Product Spaces. Fourier Transform. References. Index.

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Book SynopsisAt the same time, the author has managed to include discussions of more advanced topics such as the Gibbs phenomenon, distributions, Sturm-Liouville theory, Cesaro summability and multi-dimensional Fourier analysis, topics which one usually does not find in books at this level.Trade ReviewFrom the reviews: "This book is one in the Graduate Texts in Mathematics series published by Springer. … There is a variety of worked examples as well as 350-plus exercises … . The book is a valuable addition to the literature on Fourier analysis. It is written with more mathematical rigour than many texts … without being totally opaque to the non-specialist. … The examples at the end of each chapter are well structured and a reader working through most of them will achieve a good understanding of the topics." (Graham Brindley, The Mathematical Gazette, Vol. 90 (517), 2006) "The author … presents the results of his experiences and choices for decades of teaching courses. … The tables and formulas collected … are of great service. At the end of each chapter there is a summary section that discusses the results, gives some history, and suggests instructive exercises. We thus have a solid course on Fourier analysis and its applications interesting for students and specialists in engineering as well as for mathematicians. … I believe that the book will find numerous interested readers." (Elijah Liflyand, Zentralblatt MATH, Vol. 1032 (7), 2004) "This book is an interesting mixture of a traditional approach … and a more modern one, emphasizing the role of (tempered) distributions and the application aspects of Fourier analysis. … The book is certainly highly recommendable for those who want to learn the essence of Fourier analysis in a mathematically correct way without having to go through too much technical details." (H.G. Feichtinger, Monatshefte für Mathematik, Vol. 143 (2), 2004) "The book is appropriate for an advanced undergraduate or a master’s level one-term introductory course on Fourier series with applications to boundary value problems. … a deep idea is presented in a non-rigorous way both to show the usefulness of the idea and to stimulate interest in further study. … The book has a good collection of exercises … . Each chapter ends with both a summary of its main results and methods and historical notes." (Colin C. Graham, Mathematical Reviews, Issue 2004 e)Table of ContentsIntroduction * Preparations * Laplace and Z Transforms * Fourier Series * L^2 Theory * Separation of Variables * Fourier Transforms * Distributions * Multi-Dimentional Fourier Analysis * Appendix A: The ubiquitous convolution * Appendix B: The Discrete Fourier Transform * Appendix C: Formulae * Appendix D: Answers to exercises * Appendix E: Literature

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Book SynopsisIn the library at Trinity College, Cambridge in 1976, George Andrews of Pennsylvania State University discovered a sheaf of pages in the handwriting of Srinivasa Ramanujan. Soon designated as "Ramanujan’s Lost Notebook," it contains considerable material on mock theta functions and undoubtedly dates from the last year of Ramanujan’s life.Trade Reviewhematicians interested in the work of Ramanujan, will delight in studying this book … ." (Andrew V. Sills, Mathematical Reviews, Issue 2005 m)Table of ContentsPreface.- Introduction.- The Rogers–Ramanujan Continued Fraction and Its Modular Properties.- Explicit Evaluations of the Rogers–Ramanujan Continued Fraction.- A Fragment on the Rogers–Ramanujan and Cubic Continued Fractions.- The Rogers–Ramanujan Continued Fraction and Its Connections with Partitions and Lambert Series.- Finite Rogers–Ramanujan Continued Fractions.- Other q-continued Fractions.- Asymptotic Formulas for Continued Fractions.- Ramanujan’s Continued Fraction for (q2; q3)8/(q; q3)8.- The Rogers–Fine Identity.- An Empirical Study of the Rogers–Ramanujan Identities.- Rogers–Ramanujan–Slater Type Identities.- Partial Fractions.- Hadamard Products for Two q-Series.- Integrals of Theta-functions.- Incomplete Elliptic Integrals.- Infinite Integrals of q-Products.- Modular Equations in Ramanujan’s Lost Notebook.- Fragments on Lambert Series.- Location Guide.- Provenance.- References.- Index.

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Book SynopsisIt can be used for a course that provides an introduction to basic functional analysis, approximation theory, and numerical analysis, while building upon and applying basic techniques of real variable theory.Trade ReviewSecond Edition S.C. Brenner and L.R. Scott The Mathematical Theory of Finite Element Methods "[This is] a well-written book. A great deal of material is covered, and students who have taken the trouble to master at least some of the advanced material in the later chapters would be well placed to embark on research in the area." ZENTRALBLATT MATH From the reviews of the third edition: "An excelent survey of the deep mathematical roots of finite element methods as well as of some of the newest and most formal results concerning these methods. … The approach remains very clear and precise … . A significant number of examples and exercises improve considerably the accessability of the text. The authors also point out different ways the book could be used in various courses. … valuable reference and source for researchers (mainly mathematicians) in the topic." (Calin Ioan Gheorghiu, Zentralblatt MATH, Vol. 1135 (13), 2008)Table of ContentsPreface(3rdEd).- Preface(2ndEd).- Preface(1stED).- Basic Concepts.- Sobolev Spaces.- Variational Formulation of Elliptic Boundary Value Problems.- The Construction of a Finite Element of Space.- Polynomial Approximation Theory in Sobolev Spaces.- n-Dimensional Variational Problems.- Finite Element Multigrid Methods.- Additive Schwarz Preconditioners.- Max-norm Estimates.- Adaptive Meshes.- Variational Crimes.- Applications to Planar Elasticity.- Mixed Methods.- Iterative Techniques for Mixed Methods.- Applications of Operator-Interpolation Theory.- References.- Index.

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Book SynopsisI Youth.- II Friends and Teachers.- III Doctor of Philosophy.- IV Paris.- V Gordan's Problem.- VI Changes.- VII Only Number Fields.- VIII Tables, Chairs, and Beer Mugs.- IX Problems.- X The Future of Mathematics.- XI The New Century.- XII Second Youth.- XIII The Passionate Scientific Life.- XIV Space, Time and Number.- XV Friends and Students.- XVI Physics.- XVII War.- XVIII The Foundations of Mathematics.- XIX The New Order.- XX The Infinite!.- XXI Borrowed Time.- XXII Logic and the Understanding of Nature.- XXIII Exodus.- XXIV Age.- XXV The Last Word.Trade ReviewFrom the reviews: THE BULLETIN OF MATHEMATICS BOOKS "Originally published to great acclaim, both books explore the dramatic scientific history expressed in the lives of these two great scientists and described in the lively, nontechnical writing style of Constance Reid."Table of ContentsI Youth.- II Friends and Teachers.- III Doctor of Philosophy.- IV Paris.- V Gordan’s Problem.- VI Changes.- VII Only Number Fields.- VIII Tables, Chairs, and Beer Mugs.- IX Problems.- X The Future of Mathematics.- XI The New Century.- XII Second Youth.- XIII The Passionate Scientific Life.- XIV Space, Time and Number.- XV Friends and Students.- XVI Physics.- XVII War.- XVIII The Foundations of Mathematics.- XIX The New Order.- XX The Infinite!.- XXI Borrowed Time.- XXII Logic and the Understanding of Nature.- XXIII Exodus.- XXIV Age.- XXV The Last Word.

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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 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 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 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 SynopsisThis treatise deals with modern theory of functional equations in several variables and their applications to mathematics, information theory, and the natural, behavioural and social sciences. The authors have chosen to emphasize applications, though not at the expense of theory, so they have kept the prerequisites to a minimum.Trade Review"The book has been designed so that the chapters can be read almost independently of each other. This beautifully written treatise is very useful as a reference book for research workers in the area." Mathematical Reviews"...this is an excellent reference book, in general, for senior and higher level physics students, and perhaps for numerical analysts working on computer algorithms....I would not hesitate to recommend this book to any physics and geophysics graduate students, as well as to some interested faculty members." Physics in CanadaTable of ContentsPreface; Further information; 1. Axiomatic motivation of vector addition; 2. Cauchy's equation: Hamel basis; 3. Three further Cauchy equations: an application to information theory; 4. Generalizations of Cauchy's equations to several multiplace vector and matrix functions: an application to geometric objects; 5. Cauchy's equations for complex functions: applications to harmonic analysis and to information measures; 6. Conditional Cauchy equations: an application to geometry and a characterization of the Heaviside functions; 7. Addundancy, extensions, quasi-extensions and extensions almost everywhere: applications to harmonic analysis and to rational decision making; 8. D'Alembert's functional equation: an application to noneuclidean mechanics; 9. Images of sets and functional equations: applications to relativity theory and to additive functions bounded on particular sets; 10. Some applications of functional equations in functional analysis, in the geometry of Banach spaces and in valauation theory; 11. Characterizations of inner product spaces: an application to gas dynamics; 12. Some related equations and systems of equations: applications to combinatorics and Markov processes; 13. Equations for trigonometric and similar functions; 14. A class of equations generalizing d'Alembert and Cauchy Pexider-type equations; 15. A further generalization of Pexider's equation: a uniqueness theorem: an application to mean values; 16. More about conditional Cauchy equations: applications to additive number theoretical functions and to coding theory; 17. Mean values, mediality and self-distributivity; 18. Generalized mediality: connection to webs and nomograms; 19. Further composite equations: an application to averaging theory; 20. Homogeneity and some generalizations: applications to economics; 21. Historical notes; Notations and symbols; Hints to selected 'exercises and further results'; Bibliography; Author index; Subject index.

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Book SynopsisThe first book to assemble the wide body of theory which has rapidly developed on the dynamics of linear operators. Written for researchers in operator theory, but also accessible to anyone with a reasonable background in functional analysis at the graduate level.Trade Review'The beginning of each chapter sets the stage for what is to follow and each concludes with detailed comments, notes, references and a carefully selected list of problems. These serve the reader well, providing historical motivation as well as resources for further work … the book … is well developed, the material is well motivated and the selected topics that are explained include original results along with important simplifications of proofs from the existing research literature. The book has an extensive list of references covering the topics discussed, making it an excellent guide for students of the subject … this well-written book is a valuable resource for anyone working in Operator Theory, but is also accessible to anyone with a reasonable background in functional analysis at the graduate level.' Mathematical ReviewsTable of ContentsIntroduction; 1. Hypercyclic and supercyclic operators; 2. Hypercyclicity everywhere; 3. Connectedness and hypercyclicity; 4. Weakly mixing operators; 5. Ergodic theory and linear dynamics; 6. Beyond hypercyclicity; 7. Common hypercyclic vectors; 8. Hypercyclic subspaces; 9. Supercyclicity and the angle criterion; 10. Linear dynamics and the weak topology; 11. Universality of the Riemann zeta function; 12. About 'the' Read operator; Appendices; Notations; Index; Bibliography.

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Book SynopsisFirst published in 1968, An Introduction to Harmonic Analysis has firmly established itself as a classic text and a favorite for students and experts alike. This new edition has been revised by the author, to include several new sections and a new appendix.Trade Review'… the third, revised, edition … Katznelson managed to improve on a seemingly perfect book!' Nieuw Archif voor WiskundeTable of Contents1. Fourier series on T; 2. The convergence of Fourier series; 3. The conjugate function; 4. Interpolation of linear operators; 5. Lacunary series and quasi-analytic classes; 6. Fourier transforms on the line; 7. Fourier analysis on locally compact Abelian groups; 8. Commutative Banach algebras; A. Vector-valued functions; B. Probabilistic methods.

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Book SynopsisThis book's aim is to provide the reader with a basic understanding of Fourier series and transforms, and Laplace transforms. It is based on the authors' notes for a one semester course. Prerequisites are a basic course in both calculus and linear algebra. Otherwise the material is self-contained with exercises and examples of applications.Trade Review"With its frequent examples and exercises, the present book is eminently suitable for both self-study and a one-semester course." ChoiceTable of Contents1. Notation and terminology; 2. Background: inner product spaces; 3. Fourier series; 4. Fourier transforms; 5. The Laplace transform; Appendices.

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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 SynopsisNow in paperback, the main theme of this book is the study of geometric properties of general sets and measures in euclidean spaces. Applications of this theory include fractal-type objects such as strange attractors for dynamical systems and those fractals used as models in the sciences. The author provides a firm and unified foundation and develops all the necessary main tools, such as covering theorems, Hausdorff measures and their relations to Riesz capacities and Fourier transforms. The last third of the book is devoted to the Beisovich-Federer theory of rectifiable sets, which form in a sense the largest class of subsets of euclidean space posessing many of the properties of smooth surfaces. These sets have wide application including the higher-dimensional calculus of variations. Their relations to complex analysis and singular integrals are also studied. Essentially self-contained, this book is suitable for graduate students and researchers in mathematics.Trade Review"Provides a unified theory for the study of the topic and develops the main tools used in its study including theorems, Hausdorff measures, and their relations to Riesz capacities and Fourier transforms." Book News, Inc.Table of ContentsAcknowledgements; Basic notation; Introduction; 1. General measure theory; 2. Covering and differentiation; 3. Invariant measures; 4. Hausdorff measures and dimension; 5. Other measures and dimensions; 6. Density theorems for Hausdorff and packing measures; 7. Lipschitz maps; 8. Energies, capacities and subsets of finite measure; 9. Orthogonal projections; 10. Intersections with planes; 11. Local structure of s-dimensional sets and measures; 12. The Fourier transform and its applications; 13. Intersections of general sets; 14. Tangent measures and densities; 15. Rectifiable sets and approximate tangent planes; 16. Rectifiability, weak linear approximation and tangent measures; 17. Rectifiability and densities; 18. Rectifiability and orthogonal projections; 19. Rectifiability and othogonal projections; 19. Rectifiability and analytic capacity in the complex plane; 20. Rectifiability and singular intervals; References; List of notation; Index of terminology.

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Book SynopsisIntermediate in level between an advanced textbook and a monograph, this book covers both the classical and representation theoretic views of automorphic forms in a style which is accessible to graduate students entering the field. The treatment is based on complete proofs, which reveal the uniqueness principles underlying the basic constructions. The book features extensive foundational material on the representation theory of GL(1) and GL(2) over local fields, the theory of automorphic representations, L-functions and advanced topics such as the Langlands conjectures, the Weil representation, the RankinâSelberg method and the triple L-function, examining this subject matter from many different and complementary viewpoints. Researchers as well as students will find this a valuable guide to a notoriously difficult subject.Trade Review'This important textbook closes a gap in the existing literature, for it presents the 'representation theoretic' viewpoint of the theory of automorphic forms on GL(2) … it will become a stepping stone for many who want to study the Corvallis Proceedings or the Lecture Notes by H. Jaquet and R. Langlands or seek a pathway to R. Langland's conjectures.' Monatshefte für Mathematik'Students and researchers will find the book an understandable and penetrating treatment of a beautiful theory.' European Mathematical SocietyTable of Contents1. Modular forms; 2. Automorphic forms and representations of GL( 2, R); 3. Automorphic representations; 4. GL(2) over a p-adic field.

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Book SynopsisThis text covers key results in functional analysis that are essential for further study in analysis, the calculus of variations, dynamical systems, and the theory of partial differential equations. More than 200 fully-worked exercises and detailed proofs are given, making this ideal for upper undergraduate and beginning graduate courses.Trade Review'This excellent introduction to functional analysis brings the reader at a gentle pace from a rudimentary acquaintance with analysis to a command of the subject sufficient, for example, to start a rigorous study of partial differential equations. The choice and order of topics are very well thought-out, and there is a fine balance between general results and concrete examples and applications.' Charles Fefferman, Princeton University, New Jersey'An Introduction to Functional Analysis covers everything that one would expect to meet in an undergraduate course on this elegant area and more, including spectral theory, the category-based theorems and unbounded operators. With a well-written narrative and clear detailed proofs, together with plentiful examples and exercises, this is both an excellent course book and a valuable reference for those encountering functional analysis from across mathematics and science.' Kenneth Falconer, University of St Andrews, Scotland'This is a beautifully written book, containing a wealth of worked examples and exercises, covering the core of the theory of Banach and Hilbert spaces. The book will be of particular interest to those wishing to learn the basic functional analytic tools for the mathematical analysis of partial differential equations and the calculus of variations.' Endre Suli, University of Oxford'… this is a valuable book. It is an accessible yet serious look at the subject, and anybody who has worked through it will be rewarded with a good understanding of functional analysis, and should be in a position to read more advanced books with profit.' Mark Hunacek, The Mathematical GazetteTable of ContentsPart I. Preliminaries: 1. Vector spaces and bases; 2. Metric spaces; Part II. Normed Linear Spaces: 3. Norms and normed spaces; 4. Complete normed spaces; 5. Finite-dimensional normed spaces; 6. Spaces of continuous functions; 7. Completions and the Lebesgue spaces Lp(Ω); Part III. Hilbert Spaces: 8. Hilbert spaces; 9. Orthonormal sets and orthonormal bases for Hilbert spaces; 10. Closest points and approximation; 11. Linear maps between normed spaces; 12. Dual spaces and the Riesz representation theorem; 13. The Hilbert adjoint of a linear operator; 14. The spectrum of a bounded linear operator; 15. Compact linear operators; 16. The Hilbert–Schmidt theorem; 17. Application: Sturm–Liouville problems; Part IV. Banach Spaces: 18. Dual spaces of Banach spaces; 19. The Hahn–Banach theorem; 20. Some applications of the Hahn–Banach theorem; 21. Convex subsets of Banach spaces; 22. The principle of uniform boundedness; 23. The open mapping, inverse mapping, and closed graph theorems; 24. Spectral theory for compact operators; 25. Unbounded operators on Hilbert spaces; 26. Reflexive spaces; 27. Weak and weak-* convergence; Appendix A. Zorn's lemma; Appendix B. Lebesgue integration; Appendix C. The Banach–Alaoglu theorem; Solutions to exercises; References; Index.

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Book SynopsisAn introduction to the theory of operator spaces, emphasising examples that illustrate the theory and applications to C*-algebras, and applications to non self-adjoint operator algebras, and similarity problems. Postgraduate and professional mathematicians interested in functional analysis, operator algebras and theoretical physics will find the book has much to offer.Trade Review'The tone of the book is quite informal, friendly and inviting. Even to experts in the field, a large proportion of the results, and certainly of the proofs, will be new and stimulating. … there are literally thousands of wonderful results and insights in the text which the reader will not find elsewhere. The book covers an incredible amount of ground, and makes use of some of the most exciting recent work in modern analysis. … It is a magnificent book: an enormous treasure trove, and a work of love and care by one of the great analysts of our time. All students and researchers in functional analysis should have a copy. Anybody planning to work in operator space theory will need to be thoroughly immersed in it.' Proceedings of the Edinburgh Mathematical SocietyTable of ContentsPart I. Introduction to Operator Spaces: 1. Completely bounded maps; 2. Minimal tensor product; 3. Minimal and maximal operator space structures on a Banach space; 4. Projective tensor product; 5. The Haagerup tensor product; 6. Characterizations of operator algebras; 7. The operator Hilbert space; 8. Group C*-algebras; 9. Examples and comments; 10. Comparisons; Part II. Operator Spaces and C*-tensor products: 11. C*-norms on tensor products; 12. Nuclearity and approximation properties; 13. C*; 14. Kirchberg's theorem on decomposable maps; 15. The weak expectation property; 16. The local lifting property; 17. Exactness; 18. Local reflexivity; 19. Grothendieck's theorem for operator spaces; 20. Estimating the norms of sums of unitaries; 21. Local theory of operator spaces; 22. B(H) * B(H); 23. Completely isomorphic C*-algebras; 24. Injective and projective operator spaces; Part III. Operator Spaces and Non Self-Adjoint Operator Algebras: 25. Maximal tensor products and free products of non self-adjoint operator algebras; 26. The Blechter-Paulsen factorization; 27. Similarity problems; 28. The Sz-nagy-halmos similarity problem; Solutions to the exercises; References.

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Book SynopsisThis text is designed both for students of probability and stochastic processes, and for students of functional analysis. For the reader not familiar with functional analysis a detailed introduction to necessary notions and facts is provided. However, this is not a straight textbook in functional analysis; rather, it presents some chosen parts of functional analysis that can help understand ideas from probability and stochastic processes. The subjects range from basic Hilbert and Banach spaces, through weak topologies and Banach algebras, to the theory of semigroups of bounded linear operators. Numerous standard and non-standard examples and exercises make the book suitable as a course textbook or for self-study.Trade Review"My impression is that this text might well succeed as an attractive introduction to, or even as propaganda for the subject of probability and stochastic processes for a well-educated analyst without a probabilistic background." N.H. Bingham, Journal of the American Statistical AssociationTable of ContentsPreface; 1. Preliminaries, notations and conventions; 2. Basic notions in functional analysis; 3. Conditional expectation; 4. Brownian motion and Hilbert spaces; 5. Dual spaces and convergence of probability measures; 6. The Gelfand transform and its applications; 7. Semigroups of operators and Lévy processes; 8. Markov processes and semigroups of operators; 9. Appendixes; References; Index.

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Book SynopsisFirst published in 1968, An Introduction to Harmonic Analysis has firmly established itself as a classic text and a favorite for students and experts alike. This new edition has been revised by the author, to include several new sections and a new appendix.Trade Review'… the third, revised, edition … Katznelson managed to improve on a seemingly perfect book!' Nieuw Archif voor WiskundeTable of Contents1. Fourier series on T; 2. The convergence of Fourier series; 3. The conjugate function; 4. Interpolation of linear operators; 5. Lacunary series and quasi-analytic classes; 6. Fourier transforms on the line; 7. Fourier analysis on locally compact Abelian groups; 8. Commutative Banach algebras; A. Vector-valued functions; B. Probabilistic methods.

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Book SynopsisThis accessible text covers key results in functional analysis that are essential for further study in the calculus of variations, analysis, dynamical systems, and the theory of partial differential equations. The treatment of Hilbert spaces covers the topics required to prove the HilbertSchmidt theorem, including orthonormal bases, the Riesz representation theorem, and the basics of spectral theory. The material on Banach spaces and their duals includes the HahnBanach theorem, the KreinMilman theorem, and results based on the Baire category theorem, before culminating in a proof of sequential weak compactness in reflexive spaces. Arguments are presented in detail, and more than 200 fully-worked exercises are included to provide practice applying techniques and ideas beyond the major theorems. Familiarity with the basic theory of vector spaces and point-set topology is assumed, but knowledge of measure theory is not required, making this book ideal for upper undergraduate-level and begTrade Review'This excellent introduction to functional analysis brings the reader at a gentle pace from a rudimentary acquaintance with analysis to a command of the subject sufficient, for example, to start a rigorous study of partial differential equations. The choice and order of topics are very well thought-out, and there is a fine balance between general results and concrete examples and applications.' Charles Fefferman, Princeton University, New Jersey'An Introduction to Functional Analysis covers everything that one would expect to meet in an undergraduate course on this elegant area and more, including spectral theory, the category-based theorems and unbounded operators. With a well-written narrative and clear detailed proofs, together with plentiful examples and exercises, this is both an excellent course book and a valuable reference for those encountering functional analysis from across mathematics and science.' Kenneth Falconer, University of St Andrews, Scotland'This is a beautifully written book, containing a wealth of worked examples and exercises, covering the core of the theory of Banach and Hilbert spaces. The book will be of particular interest to those wishing to learn the basic functional analytic tools for the mathematical analysis of partial differential equations and the calculus of variations.' Endre Suli, University of Oxford'… this is a valuable book. It is an accessible yet serious look at the subject, and anybody who has worked through it will be rewarded with a good understanding of functional analysis, and should be in a position to read more advanced books with profit.' Mark Hunacek, The Mathematical GazetteTable of ContentsPart I. Preliminaries: 1. Vector spaces and bases; 2. Metric spaces; Part II. Normed Linear Spaces: 3. Norms and normed spaces; 4. Complete normed spaces; 5. Finite-dimensional normed spaces; 6. Spaces of continuous functions; 7. Completions and the Lebesgue spaces Lp(Ω); Part III. Hilbert Spaces: 8. Hilbert spaces; 9. Orthonormal sets and orthonormal bases for Hilbert spaces; 10. Closest points and approximation; 11. Linear maps between normed spaces; 12. Dual spaces and the Riesz representation theorem; 13. The Hilbert adjoint of a linear operator; 14. The spectrum of a bounded linear operator; 15. Compact linear operators; 16. The Hilbert–Schmidt theorem; 17. Application: Sturm–Liouville problems; Part IV. Banach Spaces: 18. Dual spaces of Banach spaces; 19. The Hahn–Banach theorem; 20. Some applications of the Hahn–Banach theorem; 21. Convex subsets of Banach spaces; 22. The principle of uniform boundedness; 23. The open mapping, inverse mapping, and closed graph theorems; 24. Spectral theory for compact operators; 25. Unbounded operators on Hilbert spaces; 26. Reflexive spaces; 27. Weak and weak-* convergence; Appendix A. Zorn's lemma; Appendix B. Lebesgue integration; Appendix C. The Banach–Alaoglu theorem; Solutions to exercises; References; Index.

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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 SynopsisPrinceton University's Elias Stein was the first mathematician to see the profound interconnections that tie classical Fourier analysis to several complex variables and representation theory. This volume gathers papers from internationally renowned mathematicians, many of whom have been Stein's students.Table of ContentsPreface ix Chapter 1 Selected Theorems by Eli Stein 1 Charles Fefferman Chapter 2 Eli's Impact: A Case Study 35 Charles Fefferman Chapter 3 On Oscillatory Integral Operators in Higher Dimensions 47 Jean Bourgain Chapter 4 Holder Regularity for Generalized Master Equations with Rough Kernels 63 Luis Caffarelli and Luis Silvestre Chapter 5 Extremizers of a Radon Transform Inequality 84 Michael Christ Chapter 6 Should We Solve Plateau's Problem Again? 108 Guy David Chapter 7 Averages along Polynomial Sequences in Discrete Nilpotent Lie Groups: Singular Radon Transforms 146 Alexandru D. Ionescu, Akos Magyar, and Stephen Wainger Chapter 8 Internal DLA for Cylinders 189 David Jerison, Lionel Levine, and Scott Sheffield Chapter 9 The Energy Critical Wave Equation in 3D 215 Carlos Kenig Chapter 10 On the Bounded L2 Curvature Conjecture 224 Sergiu Klainerman Chapter 11 On Div-Curl for Higher Order 245 Loredana Lanzani and Andrew S. Raich Chapter 12 Square Functions and Maximal Operators Associated with Radial Fourier Multipliers 273 Sanghyuk Lee, Keith M. Rogers, and Andreas Seeger Chapter 13 Problems of Harmonic Analysis Related to Finite-Type Hypersurfaces in 3, and Newton Polyhedra 303 Detlef Muller Chapter 14 Multi-Linear Multipliers Associated to Simplexes of Arbitrary Length 346 Camil Muscalu, Terence Tao, and Christoph Thiele Chapter 15 Diagonal Estimates for Bergman Kernels in Monomial-Type Domains 402 Alexander Nagel and Malabika Pramanik Chapter 16 On the Singularities of the Pluricomplex Green's Function 419 D. H. Phong and Jacob Sturm Chapter 17 Smoothness of Spectral Multipliers and Convolution Kernels in Nilpotent Gelfand Pairs 436 Fulvio Ricci Chapter 18 On Eigenfunction Restriction Estimates and L4-Bounds for Compact Surfaces with Nonpositive Curvature 447 Christopher D. Sogge and Steve Zelditch List of Contributors 463 Index 465

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Book Synopsis1 Introduction to the Locally Trivial Bundles Theory.- 2 Homotopy Invariants of Vector Bundles.- 3 Geometric Constructions of Bundles.- 4 Calculation Methods in K-Theory.- 5 Elliptic Operators on Smooth Manifolds and K-Theory.- 6 Some Applications of Vector Bundle Theory.- References.Table of ContentsPreface. 1. Introduction to the Locally Trivial Bundles Theory. 2. Homotopy Invariants of Vector Bundles. 3. Geometric Constructions of Bundles. 4. Calculation Methods in K-Theory. 5. Elliptic Operators on Smooth Manifolds and K-Theory. 6. Some Applications of Vector Bundle Theory. Index. References.

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Book Synopsis1 Regularizing Algorithms for Linear Ill-Posed Problems: Unified Approach.- 2 Iteration Steepest Descent Methods for Linear Operator Equations.- 3 Iteration Conjugate Direction Methods for Linear Operator Equations.- 4 Iteration Steepest Descent Methods for Nonlinear Operator Equations.- 5 Iteration Methods for Ill-Posed Constrained Minimization Problems.- 6 Descriptive Regularization Algorithms on the Basis of the Conjugate Gradient Projection Method.Trade Review`The book will be useful for specialists who in their theoretical and applied investigations deal with ill-posed and inverse problems.' Mathematical Reviews Clippings (2001)Table of ContentsPreface. Introduction. 1. Regularizing Algorithms for Linear Ill-Posed Problems: Unified Approach. 2. Iteration Steepest Descent Methods for Linear Operator Equations. 3. Iteration Conjugate Direction Methods For Linear Operator Equations. 4. Iteration Steepest Descent Methods for Nonlinear Operator Equations. 5. Iteration Methods for Ill-Posed Constrained Minimization Problems. 6. Descriptive Regularization Algorithms on the Basis of the Conjugate Gradient Projection Method. Bibliography. Index.

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Book Synopsisrii application of linear operators on a Hilbert space. We begin with a chapter on the geometry of Hilbert space and then proceed to the spectral theory of compact self adjoint operators;Table of ContentsIntroduction * I. Hilbert Spaces * II. Bounded Linear Operators on Hilbert Spaces * III. Spectral Theory of Compact Self Adjoint Operators * IV. Spectral Theory of Integral Operators * V. Oscillations of an Elastic String * VI. Operational Calculus with Applications * VII. Solving Linear Equations by Iterative Methods * VIII. Further Developments of the Spectral Theorem * IX. Banach Spaces * X. Linear Operators on a Banach Space * XI. Compact Operators on a Banach Space * XII. Non-Linear Operators * Appendix 1. Countable Sets and Separable Hilbert Spaces * Appendix 2. Lebesgue Integration and LP Spaces * Appendix 3. Proof of the Hahn-Banach Theorem * Appendix 4. Proof of the Closed Graph Theorem * Suggested Reading * References * Index

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Book SynopsisIn particular, if f and j are x2 x 2 2 2 both O(e- / ), then f = j = Ae- / , where A is a constant;Trade Review"This nicely written book by Thangavelu is concerned with extensions of Hardy's theorem to settings that arise from noncommutative harmonic analysis.... Each chapter contains several applications to the heat equation in various settings and ends with an extensive presentation of related topics, current research, detailed references to the literature, and lists of open problems. This makes the book an invaluable resource for graduate students and researchers in harmonic analysis and applied mathematics." —SIAM Review "…Each chapter ends with useful notes and open problems. Everything is written in sufficient detail to benefit specialized interested readers…" —MATHEMATICAL REVIEWS "The authoer discusses inthe present book the original theorem of Hardy and some of its generaliztions and its connections to noncommunitave analysis, harmonic analysis and special functions. First Hardy's theorem for the Euclidian Fourier transform is treated, and a theorem of Beurling and Hömander Subsequently Hardy's theorem is dicussed for the Fourier transfom on the Heisenberg group. finally the author discusses generaliztions of Hardy's theorem involving the Helgason Fourier transform for rank one symmetric spaces and for H-type groups. This unique book will be of great value for readers interested in this branch of analysis." ---Monatshefte für MathematikTable of Contents1 Euclidean Spaces.- 1.1 Fourier transform on L1(?n).- 1.2 Hermite functions and L2 theory.- 1.3 Spherical harmonics and symmetry properties.- 1.4 Hardy’s theorem on ?n.- 1.5 Beurling’s theorem and its consequences.- 1.6 Further results and open problems.- 2 Heisenberg Groups.- 2.1 Heisenberg group and its representations.- 2.2 Fourier transform on Hn.- 2.3 Special Hermite functions.- 2.4 Fourier transform of radial functions.- 2.5 Unitary group and spherical harmonics.- 2.6 Spherical harmonics and the Weyl transform.- 2.7 Weyl correspondence of polynomials.- 2.8 Heat kernel for the sublaplacian.- 2.9 Hardy’s theorem for the Heisenberg group.- 2.10 Further results and open problems.- 3 Symmetric Spaces of Rank 1.- 3.1 A Riemannian space associated to Hn.- 3.2 The algebra of radial functions on S.- 3.3 Spherical Fourier transform.- 3.4 Helgason Fourier transform.- 3.5 Hecke-Bochner formula for the Helgason Fourier transform.- 3.6 Jacobi transforms.- 3.7 Estimating the heat kernel.- 3.8 Hardy’s theorem for the Helgason Fourier transform.- 3.9 Further results and open problems.

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Book SynopsisThe treatment emphasizes applications that relate distributions to linear partial differential equations and Fourier analysis problems found in mechanics, optics, quantum mechanics, quantum field theory, and signal analysis.Trade ReviewFrom the reviews:“This is a fantastic book…Even in this enlightened age, the theory of distributions is a highly misunderstood and undervalued business…All of this notwithstanding, learning it from the extant standard sources has always been a rather an austere affair…Distributions: Theory and Applications is laden with examples and exercises…[The book] comes equipped with a long section containing solutions to selected exercises. Distributions: Theory and Applications is much more than a textbook for a one-semester introduction to the indicated subject: the reader/student who hangs in, reads the text while filling the margins and a notebook or two (or three, or four) along the way, and wrestles with the exercises, will be more than ready for more advanced works in the area as well as the business of applying distributions in a number of nontrivial settings. Additionally he will have been exposed to a great deal of serious mathematics from, if you’ll pardon the pun, all over the spectrum.”BLL — The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition. —MAA Reviews“[Distributions: Theory and Applications] is a very useful, well-written, self contained, motivating book presenting the essentials of the theory of distributions of Schwartz, together with many applications to different areas of mathematics, like linear partial differential equations, Fourier analysis, quantum mechanics and signal analysis…One of the main features of this book is that many clarifying examples, presented with full detail, are included in the text. Moreover, a large number of problems is included at the end of each chapter…This reviewer found the book under review very stimulating, informative and inviting to further study. It is a very useful text for students who want to learn the theory of distributions, for experts in the abstract theory who want to read an application-oriented presentation with many examples and, last but not least, for mathematicians in related areas or theoretical physicists who need a self-contained, reasonable short presentation of the theory. I would strongly recommend it to all my colleagues.” —Zentralblatt MATH “The aim of this book is to present the theory of distributions of Laurent Schwartz in a rigorous, accessible way, together with applications to linear partial differential equations, Fourier analysis, quantum mechanics and signal analysis. The text of Duistermaat and Kolk emphasizes applications to natural sciences like optics, quantum field theory, signal reconstruction or computer tomography. The exposition stresses applications and interactions with other parts of mathematics...It is a useful text for students who want to learn the theory of distributions, for experts in the abstract theory who want to read an application-oriented presentation with many examples, and also for mathematicians in related areas or theoretical physicists who need a self-contained presentation of the theory. The book includes many clarifying examples, presented in full detail. Many exercises and problems, with different levels of difficulty, are included at the end of each chapter. A reader who tries to solve them will re-examine many aspects of analysis of several variables in the light of the theory of distributions. Complete solutions of 146 of the 281 problems are provided and hints are given for many others.In summary, this is a useful, well-written, self-contained book about the theory of distributions of Schwartz and applications to different areas of mathematics and physics. Although there are many good books that present an introduction to the theory of distributions of Schwartz, assuming different levels of knowledge of linear functional analysis, the text of Duistermaat and Kolk is a very welcome addition. In the opinion of this reviewer, this is a stimulating, informative book that invites further study, and I strongly recommend it to any readers interested in this topic.” —Mathematical Reviews “This is a wonderful graduate level book providing a modern introduction to distribution theory with several entrance points for deeper tours into advanced analysis and applications. The lucid discussions of the development of certain key notions and the fine careful proofs define it as a valuable text book for a corresponding university course. … even in independent self-study the reader has a fair chance to obtain a reasonable amount of working knowledge … .” (G. Hörmann, Monatshefte für Mathematik, Vol. 163 (1), May, 2011)“The authors have set out to produce a text on distribution theory accessible to undergraduate (or beginning graduate) students at the point in their studies where it could serve as an alternative to a beginning course in measure theory. … The authors discuss many fascinating results in distribution theory, with an emphasis on the use of the subject in linear PDEs … . The book succeeds both as a basic and as a rich account of results in distribution theory.” (Michael Taylor, SIAM Review, Vol. 54 (3), 2012)Table of ContentsPreface.- Standard Notation.- 1 Motivation .- Problems.- 2 Test Functions.- Problems.- 3 Distributions.- Problems.- 4 Differentiation of Distributions.- Problems.- 5 Convergence of Distributions.- Problems.- 6 Taylor Expansion in Several Variables.- Problems.- 7 Localization.- Problems.- 8 Distributions with Compact Support.- Problems.- 9 Multiplication by Functions.- Problems.- 10 Transposition: Pullback and Pushforward.- Problems.- 11 Convolution of Distributions.- Problems.- 12 Fundamental Solutions.- Problems.- 13 Fractional Integration and Differentiation .- 13.1 The Case of Dimension One.- 13.2 Wave Family.- 13.3 Appendix: Euler’s Gamma Function.- Problems.- 14 Fourier Transform.- Problems.- 15 Distribution Kernels.- Problems.- 16 Fourier Series.- Problems.- 17 Fundamental Solutions and Fourier Transform.- 17.1 Appendix: Fundamental Solution of .I?/k.- Problems.- 18 Supports and Fourier Transform.- Problems.- 19 Sobolev Spaces.- Problems.- 20 Appendix: Integration.- 21 Solutions to Selected Problems.- References.- Index of Notation.- Index.

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Book SynopsisCovering the fundamental underlying theory as well as a range of applications, this unique text provides a unified overview of reproducing kernel Hilbert spaces. It offers an unrivalled and accessible introduction to the field, ideal for graduate students and researchers working in functional analysis or its applications.Trade Review'The purpose of this fine monograph is two-fold. On the one hand, the authors introduce a wide audience to the basic theory of reproducing kernel Hilbert spaces (RKHS), on the other hand they present applications of this theory in a variety of areas of mathematics … the authors have succeeded in arranging a very readable modern presentation of RKHS and in conveying the relevance of this beautiful theory by many examples and applications.' Dirk Werner, Zentralblatt MATH'Anyone looking for a nice introduction to this theory need look no further.' Jeff Ibbotson, MAA ReviewsTable of ContentsPart I. General Theory: 1. Introduction; 2. Fundamental results; 3. Interpolation and approximation; 4. Cholesky and Schur; 5. Operations on kernels; 6. Vector-valued spaces; Part II. Applications and Examples: 7. Power series on balls and pull-backs; 8. Statistics and machine learning; 9. Negative definite functions; 10. Positive definite functions on groups; 11. Applications of RKHS to integral operators; 12. Stochastic processes.

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Book SynopsisG. H. Hardy ranks among the great mathematicians of the twentieth century, doing essential research in number theory and analysis. This book is a feast of Hardy's writing, featuring articles ranging from the serious to the humorous. The G. H. Hardy Reader is a worthy introduction to an extraordinary individual.Trade Review'The editors are to be congratulated on putting together this beautiful 'reader' with material from so many different sources, which illustrates so well the life, character and work of one of the great mathematicians of the twentieth century, Godfrey Harold Hardy (1877-1947). Even if you are familiar with Hardy's masterpiece A Mathematician's Apology or his book on Ramanujan, Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work you will find a wealth of new and fascinating material in this 'reader' about Hardy.' Kenneth S. Williams, Canadian Mathematical Society NotesTable of ContentsPart I. Biography: 1. Hardy's life; 2. The letter from Ramanujan to Hardy, 16 January 1913; 3. A letter from Bertrand Russell to Lady Ottoline Morrell, 2 February 1913; 4. The Indian mathematician Ramanujan; 5. Epilogue from the man who knew infinity; 6. Posters of 'Hardy's years at Oxford'; 7. A glimpse of J. E. Littlewood; 8. A letter from Freeman Dyson to C. P. Snow, 22 May 1967, and two letters from Hardy to Dyson; 9. Miss Gertrude Hardy; Part II. Writings by and about G. H. Hardy: 10. Hardy on writing books; 11. Selections from Hardy's writings; 12. Selections from what others have said about Hardy; Part III. Mathematics: 13. An introduction to the theory of numbers; 14. Prime numbers; 15. The theory of numbers; 16. The Riemann zeta-function and lattice point problems; 17. Four Hardy gems; 18. What is geometry?; 19. The case against the mathematical tripos; 20. The mathematician on cricket; 21. Cricket for the rest of us; 22. A mathematical theorem about golf; 23. Mathematics in war-time; 24. Mathematics; 25. Asymptotic formulæ in combinatory analysis (excerpts) with S. Ramanujan; 26. A new solution of Waring's problem (excerpts), with J. E. Littlewood; 27. Some notes on certain theorems in higher trigonometry; 28. The Integral _∞0sin xx dx and further remarks on the integral _∞0sin xx dx; Part IV. Tributes: 29. Dr. Glaisher and the 'messenger of mathematics'; 30. David Hilbert; 31. Edmund Landau (with H. Heilbronn); 32. Gösta Mittag-Leffler; Part V. Book Reviews: 33. Osgood's calculus and Johnson's calculus; 34. Hadamard: the psychology of invention in the mathematical field; 35. Hulburt: differential and integral calculus; 36. Bôcher: an introduction to the study of integral equations.

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Book SynopsisThis two-volume text provides a complete overview of the theory of Banach spaces, emphasising its interplay with classical and harmonic analysis (particularly Sidon sets) and probability. The authors give a full exposition of all results, as well as numerous exercises and comments to complement the text and aid graduate students in functional analysis. The book will also be an invaluable reference volume for researchers in analysis. Volume 1 covers the basics of Banach space theory, operatory theory in Banach spaces, harmonic analysis and probability. The authors also provide an annex devoted to compact Abelian groups. Volume 2 focuses on applications of the tools presented in the first volume, including Dvoretzky's theorem, spaces without the approximation property, Gaussian processes, and more. Four leading experts also provide surveys outlining major developments in the field since the publication of the original French edition.Trade ReviewReview of previous edition: 'Undoubtedly, the book will be very useful for all mathematicians (not only for postgraduate students) who work in the theory of Banach spaces, harmonic analysis and probability theory.' Anatolij M. Plichko, American Mathematical SocietyReview from previous edition: '… carefully written and edited … The exposition is clear, precise and lively, and the text makes very good reading.' Eve Oja, Zentralblatt MathTable of ContentsPreface; 1. Euclidean sections; 2. Separable Banach spaces without the approximation property; 3. Gaussian processes; 4. Reflexive subspaces of L1; 5. The method of selectors. Examples of its use; 6. The Pisier space of almost surely continuous functions. Applications; Appendix. News in the theory of infinite-dimensional Banach spaces in the past twenty years G. Godefroy; An update on some problems in high dimensional convex geometry and related probabilistic results O. Guédon; A few updates and pointers G. Pisier; On the mesh condition for Sidon sets L. Rodriguez-Piazza; Bibliography; Author index; Notation index; Subject index.

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Book SynopsisOriginally published in 1910 as number twelve in the Cambridge Tracts in Mathematics and Mathematical Physics series, this book provides an up-to-date version of Du Bois-Reymond's InfinitÃrcalcÃl by the celebrated English mathematician G. H. Hardy. This tract will be of value to anyone with an interest in the history of mathematics or the theory of functions.Table of Contents1. Introduction; 2. Scales of infinity in general; 3. Logarithmico-exponential scales; 4. Special problems connected with logarithmico-exponential scales; 5. Functions which do not conform to any logarithmico-exponential scale; 6. Differentiation and integration; 7. Some developments of Du Bois-Reymond's Infinitärcalcül; Appendix 1. General bibliography; Appendix 2. A sketch of some applications, with references; Appendix 3. Some numerical results.

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Book SynopsisAims to provides applications to Visintin and Reshetnyak type theorems (Chapters 6 and 8), existence of solutions to differential inclusions (Chapter 7), dynamical programming (Chapter 8) and the Central Limit Theorem in locally convex spaces (Chapter 9).Trade ReviewFrom the reviews: "This book presents a wealth of results on Young measures on topological spaces in a very general framework. It is very likely that it will become the reference and starting point for any further developments in the field." (Georg K. Dolzmann, Mathematical Reviews, 2005k)Table of ContentsPreface. Generalities, Preliminary results. Young Measures, the four Stable Topologies: S, M, N, W. Convergence in Probability of Young Measures (with some applications to stable convergence). Compactness. Strong Tightness. Young Measures on Banach Spaces. Application. Applications in Control Theory. Semicontinuity of Integral Functionals using Young Measures. Stable Convergence in Limit Theorems of Probability Theory.

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Book SynopsisAt the same time, the author has managed to include discussions of more advanced topics such as the Gibbs phenomenon, distributions, Sturm-Liouville theory, Cesaro summability and multi-dimensional Fourier analysis, topics which one usually does not find in books at this level.Trade ReviewFrom the reviews: "This book is one in the Graduate Texts in Mathematics series published by Springer. … There is a variety of worked examples as well as 350-plus exercises … . The book is a valuable addition to the literature on Fourier analysis. It is written with more mathematical rigour than many texts … without being totally opaque to the non-specialist. … The examples at the end of each chapter are well structured and a reader working through most of them will achieve a good understanding of the topics." (Graham Brindley, The Mathematical Gazette, Vol. 90 (517), 2006) "The author … presents the results of his experiences and choices for decades of teaching courses. … The tables and formulas collected … are of great service. At the end of each chapter there is a summary section that discusses the results, gives some history, and suggests instructive exercises. We thus have a solid course on Fourier analysis and its applications interesting for students and specialists in engineering as well as for mathematicians. … I believe that the book will find numerous interested readers." (Elijah Liflyand, Zentralblatt MATH, Vol. 1032 (7), 2004) "This book is an interesting mixture of a traditional approach … and a more modern one, emphasizing the role of (tempered) distributions and the application aspects of Fourier analysis. … The book is certainly highly recommendable for those who want to learn the essence of Fourier analysis in a mathematically correct way without having to go through too much technical details." (H.G. Feichtinger, Monatshefte für Mathematik, Vol. 143 (2), 2004) "The book is appropriate for an advanced undergraduate or a master’s level one-term introductory course on Fourier series with applications to boundary value problems. … a deep idea is presented in a non-rigorous way both to show the usefulness of the idea and to stimulate interest in further study. … The book has a good collection of exercises … . Each chapter ends with both a summary of its main results and methods and historical notes." (Colin C. Graham, Mathematical Reviews, Issue 2004 e)Table of ContentsIntroduction * Preparations * Laplace and Z Transforms * Fourier Series * L^2 Theory * Separation of Variables * Fourier Transforms * Distributions * Multi-Dimentional Fourier Analysis * Appendix A: The ubiquitous convolution * Appendix B: The Discrete Fourier Transform * Appendix C: Formulae * Appendix D: Answers to exercises * Appendix E: Literature

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Book SynopsisConcrete Functional Calculus focuses primarily on differentiability of some nonlinear operators on functions or pairs of functions.Trade ReviewFrom the reviews:“This monograph is a thorough and masterful work on non-linear analysis designed to be read and studied by graduate students and professional mathematical researchers. The overall perspective and choice of material is highly novel and original. … It is a unique account of some key areas of modern analysis which will surely turn out to be invaluable for many researchers in this and related areas.” (David Applebaum, The Mathematical Gazette, Vol. 98 (541), March, 2014)“The present monograph is quite extensive and interesting. It is divided into twelve chapters on different topics on Functional calculus and an appendix on non-atomic measure spaces. … The book has many historical comments and remarks which clarify the developments of the theory. It has also an extensive bibliography with 258 references. … will be very useful for all interested readers in Real-Functional Analysis and Probability.” (Francisco L. Hernandez, The European Mathematical Society, January, 2012)“The monograph under review aims at analyzing properties such as Hölder continuity, differentiability and analyticity of various types of nonlinear operators which arises in the study of differential and integral equations and in applications to problems of statistics and probability. … this is an interesting book which contains a lot of material.” (Massimo Lanza de Cristoforis, Mathematical Reviews, Issue 2012 e)Table of ContentsPreface.- 1 Introduction and Overview.- 2 Definitions and Basic Properties of Extended Riemann-Stieltjes integrals.- 3 Phi-variation and p-variation; Inequalities for Integrals.- 4 Banach Algebras.- 5 Derivatives and Analyticity in Normed Spaces.- 6 Nemytskii Operators on Function Spaces.- 7 Nemytskii Oerators on Lp Spaces.- 8 Two-Function Composition.- 9 Product Integration.- 10 Nonlinear Differential and Integral Equations.- 11 Fourier Series.- 12 Stochastic Processes and Phi-Variation.- Appendix Nonatomic Measure Spaces.- References.- Subject Index.- Author Index.- Index of Notation.

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Book SynopsisPreface.- Chapter 1. Preliminaries.- Chapter 2. Fock Spaces.- Chapter 3. The Berezin Transform and BMO.- Chapter 4. Interpolating and Sampling Sequences.- Chapter 5. Zero Sets for Fock Spaces.- Chapter 6. Toeplitz Operators.- Chapter 7. Small Hankel Operators.- Chapter 8. Hankel Operators.- References.- Index.Trade ReviewFrom the reviews:“Excellent books exist in the literature on the theory of Hardy spaces … but no textbook concerning the theory of Fock spaces has appeared before. The purpose of the author is to fill this gap and provide to any researcher in the field or graduate students the appropriate place to find the results or the bibliographical references needed for their use. … author succeeds with his goal. … a great addition to the literature and in the future will become a classic in the field.” (Jordi Pau, Mathematical Reviews, January, 2013)“This book is intended to provide a convenient reference to Fock spaces. … Each chapter ends with a series of exercises. The material is presented in a pedagogical way. The reference list contains 259 relevant items. This book is well written and it is a good reference for graduate students who are interested in Fock spaces.” (Atsushi Yamamori, Zentralblatt MATH, Vol. 1262, 2013)Table of ContentsPreface.- Chapter 1. Preliminaries.- Chapter 2. Fock Spaces.- Chapter 3. The Berezin Transform and BMO.- Chapter 4. Interpolating and Sampling Sequences.- Chapter 5. Zero Sets for Fock Spaces.- Chapter 6. Toeplitz Operators.- Chapter 7. Small Hankel Operators.- Chapter 8. Hankel Operators.- References.- Index.

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Book SynopsisIn its Second Edition, this in-depth study of the vibrations of fractal strings interlinks number theory, spectral geometry and fractal geometry. Includes a geometric reformulation of the Riemann hypothesis and a new final chapter on recent topics and results.Trade Review“This interesting volume gives a thorough introduction to an active field of research and will be very valuable to graduate students and researchers alike.” (C. Baxa, Monatshefte für Mathematik, Vol. 180, 2016)“In this research monograph the authors provide a mathematical theory of complex dimensions of fractal strings and its many applications. … The book is written in a self-contained manner the results … are completely proved. I appreciate that the book is useful for mathematicians, students, researchers, postgraduates, physicians and other specialists which are interested in studying the fractals and dimension theory.” (Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, April, 2013)“The authors provide a mathematical theory of complex dimensions of fractal strings and its many applications. … The book is written in a self-contained manner, the results (including some fundamental ones) are completely proved. … the book will be useful to mathematicians, students, researchers, postgraduates, physicians and other specialists which are interested in studying fractals and dimension theory.” (Nicolae-Adrian Secelean, Zentralblatt MATH, Vol. 1261, 2013)"In this book the author encompasses a broad range of topics that connect many areas of mathematics, including fractal geometry, number theory, spectral geometry, dynamical systems, complex analysis, distribution theory and mathematical physics. The book is self containing, the material organized in chapters preceding by an introduction and finally there are some interesting applications of the theory presented. ...The book is very well written and organized and the subject is very interesting and actual and has many applications." -- Nicolae-Adrian Secelean for Zentralblatt MATH"This highly original self-contained book will appeal to geometers, fractalists, mathematical physicists and number theorists, as well as to graduate students in these fields and others interested in gaining insight into these rich areas either for its own sake or with a view to applications. They will find it a stimulating guide, well written in a clear and pleasant style." -- Mathematical Reviews (Review of previous book by authors)"It is the reviewera (TM)s opinion that the authors have succeeded in showing that the complex dimensions provide a very natural and unifying mathematical framework for investigating the oscillations in the geometry and the spectrum of a fractal string. The book is well written. The exposition is self-contained, intelligent and well paced." -- Bulletin of the London Mathematical Society (Review of previous book by authors)"The new approach and results on the important problems illuminated in this work will appeal to researchers and graduate students in number theory, fractal geometry, dynamical systems, spectral geometry, and mathematical physics." -- Simulation News Europe (Review of previous book by authors)Table of ContentsPreface.- Overview.- Introduction.- 1. Complex Dimensions of Ordinary Fractal Strings.- 2. Complex Dimensions of Self-Similar Fractal Strings.- 3. Complex Dimensions of Nonlattice Self-Similar Strings.- 4. Generalized Fractal Strings Viewed as Measures.- 5. Explicit Formulas for Generalized Fractal Strings.- 6. The Geometry and the Spectrum of Fractal Strings.- 7. Periodic Orbits of Self-Similar Flows.- 8. Fractal Tube Formulas.- 9. Riemann Hypothesis and Inverse Spectral Problems.- 10. Generalized Cantor Strings and their Oscillations.- 11. Critical Zero of Zeta Functions.- 12 Fractality and Complex Dimensions.- 13. Recent Results and Perspectives.- Appendix A. Zeta Functions in Number Theory.- Appendix B. Zeta Functions of Laplacians and Spectral Asymptotics.- Appendix C. An Application of Nevanlinna Theory.- Bibliography.- Author Index.- Subject Index.- Index of Symbols.- Conventions.- Acknowledgements.

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Book SynopsisIts discovery has frequently been deemed the mathematical equivalent of finding Beethoven's tenth symphony.This volume is the fourth of five volumes that the authors plan to write on Ramanujan’s lost notebook.​ In contrast to the first three books on Ramanujan's Lost Notebook, the fourth book does not focus on q-series.Table of ContentsPreface.- ​​​1 Introduction.- 2 Double Series of Bessel Functions and the Circle and Divisor Problems.- 3 Koshliakov's Formula and Guinand's Formula.- 4 Theorems Featuring the Gamma Function.- 5 Hypergeometric Series.- 6 Euler's Constant.- 7 Problems in Diophantine Approximation.- 8 Number Theory.- 9 Divisor Sums.- 10 Identities Related to the Riemann Zeta Function and Periodic Zeta Functions.- 11 Two Partial Unpublished Manuscripts on Sums Involving Primes.- 12 Infinite Series.- 13 A Partial Manuscript on Fourier and Laplace Transforms.- 14 Integral Analogues of Theta Functions adn Gauss Sums.- 15 Functional Equations for Products of Mellin Transforms.- 16 Infinite Products.- 17 A Preliminary Version of Ramanujan's Paper, On the Integral ∫_0^x tan^(-1)t)/t dt.- 18 A Partial Manuscript Connected with Ramanujan's Paper, Some Definite Integrals.- 19 Miscellaneous Results in Analysis.- 20 Elementary Results.- 21 A Strange, Enigmatic Partial Manuscript.- Location Guide.- Provenance.- References.- Index.

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