During the of Fall 1991, The Centre de Recerca Matematica, a research institute sponsored by the Institut d'Estudis Catalans, devoted a quarter to the study of stochastic analysis. Prominent workers in this field visited the Center from all over the world for periods ranging from a few days to several weeks. To take advantage of the presence in Barcelona of so many special ists in stochastic analysis, we organized a workshop on the subject in Sant Feliu de Guixols (Girona) that provided an opportunity for them to ex change information and ideas about their current work. Topics discussed included: Analysis on the Wiener space, Anticipating Stochastic Calculus and its Applications, Correlation Inequalities, Stochastic Flows, Reflected Semimartingales, and others. This volume contains a refereed selection of contributions from some of the participants in this workshop. We are deeply indebted to the authors of the articles for these exposi tions of their valuable research contributions. We also would like to thank all the referees for their helpful advice in making the volume a reflection of the dynamic interchange that characterized the workshop. The success of the Seminar was due essentially to the enthusiasm and stimulating discus sions of all the participants in an informal and pleasant atmosphere. To all of them our warm gratitude."

Biane, Philippe: Non-commutative stochastic calculus.-Voiculescu, Dan-Virgil: Lectures on free probability.- Guionnet, Alice: Large random matrices: Lectures on macroscopic asymptotics. "

Crisscross, zigzag, bowtie, devil, angel, or star: which are the longest, the shortest, the strongest, and the weakest lacings? Pondering the mathematics of shoelaces, the author paints a vivid picture of the simple, beautiful, and surprising characterizations of the most common shoelace patterns. The mathematics involved is an attractive mix of combinatorics and elementary calculus. This book will be enjoyed by mathematically minded people for as long as there are shoes to lace. Burkard Polster is a well-known mathematical juggler, magician, origami expert, bubble-master, shoelace charmer, and 'Count von Count' impersonator. His previous books include ""A Geometrical Picture Book"", ""The Mathematics of Juggling"", and ""QED: Beauty in Mathematical Proof"". Want to learn more about knot theory? See ""The Knot Book"" by Colin Adams and ""Knots and Links"" by Dale Rolfsen. To read a review published in the ""Gazette of the Australian Mathematical Society"", click here.

This book is devoted to a domain of highest industrial and scienti?c interest, the complexity. The complexity understanding and management will be a main source of e?ciency and prosperity for the next decades. Complex systems areassembliesof multiple subsystemsand arecharact- ized by emergent behavior that results by nonlinear interactions among the subsystems at multiple levels of organization. Evolvability that is the ability to evolve is the method to confront and surpass the successive boundaries of complexity. Evolvability is not biological but should be considered here in the sense that the corresponding systems have, at di?erent levels, charact- istics that are naturally associated to the living systems. The signi?cance of the complexity and the phenomena of emergence are highlighted in the ?rst chapterofthe book.Theimplicationofconcepts aslevelofreality, circularity and closure for evolvable systems is evaluated. The second chapter of the book exposes the methodology to analyze and manage complex systems. The polystochastic models, PSMs, are the cons- ered mathematical tools. PSMs characterize systems emerging when several stochastic processes occurring at di?erent conditioning levels, are capable to interact with each other, resulting in qualitatively new processes and s- tems. Innovative are the higher categories approach and the introduction of apartialdi?erentialmodelfor multiple levelsmodeling.This imposes making use of appropriate notions of time, space, probabilities and entropy. Categorytheoryistheformalismcapabletooutlinethegeneralframework, shared by the functional organization of biological organisms, of cognitive systems, by the operational structure of evolvable technologies and devices and after all by the scienti?c and engineering methods

Thiscollectionofproblemsisplannedasatextbookforuniversitycoursesinthe theoryofstochasticprocessesandrelatedspecialcourses. Theproblemsinthebook haveawidespectrumofthelevelofdif cultyandcanbeusefulforreaderswith variouslevelsofmasteringinthetheoryofstochasticprocesses. Togetherwithte- nicalandillustrativeproblemsintendedforbeginners,thebookcontainsanumber ofproblemsoftheoreticalnaturethatcanbeusefulforstudentsandundergraduate studentsthatpursueadvancedstudiesinthetheoryofstochasticprocessesandits- plications. Amongothers,theimportantaimofthebookistoprovideateachingstaff anef cienttoolforpreparingseminarstudies,tests,andexamsconcerninguniversity coursesinthetheoryofstochasticprocessesandrelatedtopics. Whilecomposingthe book,theauthorshavepartiallyusedthecollectionsofproblemsinprobabilityt- ory[16,65,75,83]. Also,someexercisesandproblemsfromthemonographsand textbooks[4,9,19,22,82]wereused. Atthesametime,alargepartofourproblem bookcontainsoriginalmaterial. Thebookisorganizedasfollows. Theproblemsarecollectedintochapters,each chapterbeingdevotedtoacertaintopic. Atthebeginningofeachchapter,theth- reticalgroundsforthecorrespondingtopicaregivenbrie ytogetherwiththelistof bibliography,whichthereadercanuseinordertostudythistopicinmoredetail. For themostoftheproblems,eitherhintsorcompletesolutions(oranswers)aregiven, andsomeoftheproblemsareprovidedwithbothhintsandsolutions(answers). H- ever,theauthorsdonotrecommendthatareaderusethehintssystematically,because solvingaproblemwithoutassistanceismuchmoreusefulthanusingaready-made idea. Somestatementsthathaveaparticulartheoreticalinterestareformulatedon theoreticalgrounds,andtheirproofsareformulatedasproblemsforthereader. Such problemsaresuppliedwitheithercompletesolutionsordetailedhints. Inordertoworkwiththeproblembookef ciently,areadershouldbeacquainted withprobabilitytheory,calculus,andmeasuretheorywithinthescopeofresp- tiveuniversity courses. Standard notions, suchas random variable, measurability, independence, Lebesgue measure and integral, and so on are used without ad- tionaldiscussion. Allthenewnotionsandstatementsrequiredforsolvingthepr- lemsaregiveneitherontheoreticalgroundsorintheformulationsoftheproblems vii viii Preface straightforwardly. However,sometimesanotionisusedinthetextbeforeitsformal de nition. Forinstance,theWienerandPoissonprocessesareprocesseswithin- pendentincrementsandthusareformallyintroducedinaTheoreticalgroundsfor Chapter5,buttheseprocessesareusedwidelyintheproblemsofChapters2to4. Theauthorsrecommendthatareaderwhocomestoanunknownnotionorobject usetheIndexinorderto ndthecorrespondingformalde nition. Thesamerec- mendationconcernssomestandardabbreviationsandsymbolslistedattheendofthe book. Someproblemsinthebookformcycles:solutionstooneofthemaregrounded onstatementsofothersoronauxiliaryconstructionsdescribedinsomepreceding solutions. Sometimes,onthecontrary,itisproposedtoprovethesamestatement withindifferentproblemsusingessentiallydifferenttechniques. Theauthorsrec- mendareaderpayspeci cattentiontothesefruitfulinternallinksbetweenvarious topicsofthetheoryofstochasticprocesses. Everypartofthebookwascomposedsubstantiallybyoneauthor. Chapters1-6, and16arecomposedbyA. Kulik,Chapters7,12-15,18,and19byYu. Mishura, Chapters 8-10 by A. Pilipenko, Chapter 17 by A. Kukush, and Chapter 20 by D. Gusak. Chapter11waspreparedjointlybyD. GusakandA. Pilipenko. Atthe sametime,everyauthorhasmadeacontributiontootherpartsofthebookbyprop- ingseparateproblemsorcyclesofproblems,improvingpreliminaryversionsoft- oreticalgrounds,andeditingthe naltext. The authors would like to express their deep gratitude to M. Portenko and A. Ivanovfortheircarefulreadingofapreliminaryversionofthebookandva- ablecommentsthatledtosigni cantimprovementofthetext. Theauthorsarealso gratefultoT. Yakovenko,G. Shevchenko,O. Soloveyko, Yu. Kartashov, Yu. K- menko,A. Malenko,andN. Ryabovafortheirassistanceintranslation,preparing lesandpictures,andcomposingthesubjectindexandreferences. Thetheoryofstochasticprocessesisanextendeddiscipline,andtheauthors- derstandthattheproblembookinitscurrentformmaycausecriticalremarksfrom readers,concerningeitherthestructureofthebookorthecontentofseparatech- ters. Whilepublishingtheproblembookinitscurrentform,theauthorsareopenfor remarks,comments,andpropositions,andexpressinadvancetheirgratitudetoall theircorrespondents. Kyiv DmytroGusak December2008 AlexanderKukush AlexeyKulik YuliyaMishura AndreyPilipenko Contents 1 De?nition of stochastic process. Cylinder?-algebra, ?nite-dimensional distributions, the Kolmogorov theorem...1 Theoreticalgrounds ...1 Bibliography...3 Problems...3 Hints...7 AnswersandSolutions...9 2 Characteristics of a stochastic process. Mean and covariance functions. Characteristic functions...11 Theoreticalgrounds ...11 Bibliography...13 Problems...13 Hints...16 AnswersandSolutions...17 3 Trajectories. Modi?cations. Filtrations...21 Theoreticalgrounds ...21 Bibliography...24 Problems...24 Hints...29 AnswersandSolutions...31 4 Continuity. Differentiability. Integrability...33 Theoreticalgrounds ...33 Bibliography...34 Problems...34 Hints...38 AnswersandSolutions...40 ix x Contents 5 Stochastic processes with independent increments. Wiener and Poisson processes. Poisson point measures...

The present volume gives a systematic treatment of potential functions. It takes its origin in two courses, one elementary and one advanced, which the author has given at intervals during the last ten years, and has a two-fold purpose: first, to serve as an introduction for students whose attainments in the Calculus include some knowledge of partial derivatives and multiple and line integrals; and secondly, to provide the reader with the fundamentals of the subject, so that he may proceed immediately to the applications, or to the periodical literature of the day. It is inherent in the nature of the subject that physical intuition and illustration be appealed to freely, and this has been done. However, that the book may present sound ideals to the student, and in order also serve the mathematician, both for purposes of reference and as a basis for further developments, the proofs have been given by rigorous methods. This has led, at a number of points, to results either not found elsewhere, or not readily accessible. Thus, Chapter IV contains a proof for the general regular region of the divergence theorem (Gauss', or Green's theorem) on the reduction of volume to surface integrals. The treatment of the fundamental existence theorems in Chapter XI by means of integral equations meets squarely the difficulties incident to .the discontinuity of the kernel, and the same chapter gives an account of the most recent developments with respect to the Dirichlet problem."

On behalf of those of us who in various ways have con tributed to this volume, and on behalf of all of his colleagues, students and friends throughout the world-wide scientific com munity, we dedicate this volume to Gopinath Kallianpur as a tribute to his work and in appreciation for the insights which he has so graciously and generously offered, and continues to offer, to all of us. Stochastic Processes contains 41 articles related to and frequently influ enced by Kallianpur's work. We regret that space considerations prevented us from including contributions from his numerous colleagues (at North Carolina, lSI, Minnesota, Michigan), former students, co-authors and other eminent scientists whose work is akin to Kallianpur's. This would have taken several more volumes All articles have been refereed, and for their valuable assistance in this we thank many of the contributing authors, as well as: R. Bradley, M.H.A. Davis, R. Davis, J. Hawkins, J. Horowitz, C. Houdre, N.C. Jain, C. Ji, P. Kokoszka, T. Kurtz, K.S. Lau, W. Linde, D. Monrad, D. Stroook, D. Surgailis and S. Yakowitz. We also thank June Maxwell for editorial assistance, Peggy Ravitch for help with the production of the volume, and Lisa Brooks for secretarial assistance. Finally, we are indebted to Dr. Martin Gilchrist, the Statistics editor, and the Springer editorial board for their excellent cooperation and enthusiastic support throughout this project."

The chapters in this volume, written by international experts from different fields of mathematics, are devoted to honoring George Isac, a renowned mathematician. These contributions focus on recent developments in complementarity theory, variational principles, stability theory of functional equations, nonsmooth optimization, and several other important topics at the forefront of nonlinear analysis and optimization.

This volume contains the contributions of the participants to the Oslo- Silivri Workshop on Stochastic Analysis, held in Silivri, from July 18 to July 29, at the Nazlm Terzioglu Graduate Research Center of Istanbul University. 1994, There were three lectures: * Mathematical Theory 0/ Communication Networks by V. Anantharam, * State-Space Models 0/ the Term Structure o/Interest Rates, by D. Duffie, * Theory 0/ Capacity on the Wiener Space, by F. Hirsch. The main lectures are presented at the beginning of the volume. The contributing papers cover different domains varying from random fields to dis- tributions on infinite dimensional spaces. We would like to thank the following organizations for their financial sup- port: * VISTA, a research cooperation between the Norwegian Academy of Scineces and Letters and Den Norske Stats Oljeselskap A. S. (Statsoil). * Ecole Nationale Superieure des Telecommunications de Paris. In the summer of 1994 we lost our dear friend and colleague ALBERT BADRIKIAN. We are dedicating this volume to his memory. H. Korezlioglu, B. 0ksendal, A. S. Ustunel MATHEMATICAL THEORY OF COMMUNICATION NETWORKS VENKAT ANANTHARAM * EECS DEPARTMENT UNIVERSITY OF CALIFORNIA BERKELEY, CA 94720 [email protected] Abstract We describe so me recent advances in the mathematical theory of com- munication networks.

This book contains an introduction to three topics in stochastic control: discrete time stochastic control, i. e. , stochastic dynamic programming (Chapter 1), piecewise - terministic control problems (Chapter 3), and control of Ito diffusions (Chapter 4). The chapters include treatments of optimal stopping problems. An Appendix - calls material from elementary probability theory and gives heuristic explanations of certain more advanced tools in probability theory. The book will hopefully be of interest to students in several ?elds: economics, engineering, operations research, ?nance, business, mathematics. In economics and business administration, graduate students should readily be able to read it, and the mathematical level can be suitable for advanced undergraduates in mathem- ics and science. The prerequisites for reading the book are only a calculus course and a course in elementary probability. (Certain technical comments may demand a slightly better background. ) As this book perhaps (and hopefully) will be read by readers with widely diff- ing backgrounds, some general advice may be useful: Don't be put off if paragraphs, comments, or remarks contain material of a seemingly more technical nature that you don't understand. Just skip such material and continue reading, it will surely not be needed in order to understand the main ideas and results. The presentation avoids the use of measure theory.

This book provides a systematic and comprehensive account of asymptotic sets and functions from which a broad and useful theory emerges in the areas of optimization and variational inequalities. A variety of motivations leads mathematicians to study questions about attainment of the infimum in a minimization problem and its stability, duality and minmax theorems, convexification of sets and functions, and maximal monotone maps. For each there is the central problem of handling unbounded situations. Such problems arise in theory but also within the development of numerical methods. The book focuses on the notions of asymptotic cones and associated asymptotic functions that provide a natural and unifying framework for the resolution of these types of problems. These notions have been used largely and traditionally in convex analysis, yet these concepts play a prominent and independent role in both convex and nonconvex analysis. This book covers convex and nonconvex problems, offering detailed analysis and techniques that go beyond traditional approaches. The book will serve as a useful reference and self-contained text for researchers and graduate students in the fields of modern optimization theory and nonlinear analysis.

Quasiregular Mappings extend quasiconformal theory to the noninjective case.They give a natural and beautiful generalization of the geometric aspects ofthe theory of analytic functions of one complex variable to Euclidean n-space or, more generally, to Riemannian n-manifolds. This book is a self-contained exposition of the subject. A braod spectrum of results of both analytic and geometric character are presented, and the methods vary accordingly. The main tools are the variational integral method and the extremal length method, both of which are thoroughly developed here. Reshetnyak's basic theorem on discreteness and openness is used from the beginning, but the proof by means of variational integrals is postponed until near the end. Thus, the method of extremal length is being used at an early stage and leads, among other things, to geometric proofs of Picard-type theorems and a defect relation, which are some of the high points of the present book.

This self-contained text/reference provides a basic foundation for practitioners, researchers, and students interested in any of the diverse areas of multiscale (geo)potential theory. New mathematical methods are developed enabling the gravitational potential of a planetary body to be modeled using a continuous flow of observations from land or satellite devices. Harmonic wavelets methods are introduced, as well as fast computational schemes and various numerical test examples. Presented are multiscale approaches for numerous geoscientific problems, including geoidal determination, magnetic field reconstruction, deformation analysis, and density variation modelling With exercises at the end of each chapter, the book may be used as a textbook for graduate-level courses in geomathematics, applied mathematics, and geophysics. The work is also an up-to-date reference text for geoscientists, applied mathematicians, and engineers.

Fast technological development produces systems of ever-increasing complex ity. The demand for reliable functioning of these systems has become more and more important. Thus, there is a need for highly reliable technical devices and systems, for monitoring and controlling their functioning and for planning maintenance and corrective actions to fulfill given requirements considering eco nomic limitations. These tasks reflect the wide field of engineering activities that are accompa nied by and based on a wide range of stochastical models. The book presents the main contributions to a workshop on Stochastic Models of Reliability, Qual ity, and Safety held in Schierke near Magdeburg, Germany. This workshop was part of a series of meetings that take place every two years organized by the Society of Reliability, Quality and Safety. The basic idea of these workshops is to bring together theorists, applied statisticians, and practitioners to exchange experiences and ideas of common interest. The book contains recent results in reliability and related fields. The presentation aims at making at least a part of the results accessible to engineers."

As in the case of the two previous volumes published in 1986 and 1997, the purpose of this monograph is to focus the interplay between real (functional) analysis and stochastic analysis show their mutual benefits and advance the subjects. The presentation of each article, given as a chapter, is in a research-expository style covering the respective topics in depth. In fact, most of the details are included so that each work is essentially self contained and thus will be of use both for advanced graduate students and other researchers interested in the areas considered. Moreover, numerous new problems for future research are suggested in each chapter. The presented articles contain a substantial number of new results as well as unified and simplified accounts of previously known ones. A large part of the material cov ered is on stochastic differential equations on various structures, together with some applications. Although Brownian motion plays a key role, (semi-) martingale theory is important for a considerable extent. Moreover, noncommutative analysis and probabil ity have a prominent role in some chapters, with new ideas and results. A more detailed outline of each of the articles appears in the introduction and outline to assist readers in selecting and starting their work. All chapters have been reviewed."

Intended for a second course in stationary processes, Stationary Stochastic Processes: Theory and Applications presents the theory behind the field's widely scattered applications in engineering and science. In addition, it reviews sample function properties and spectral representations for stationary processes and fields, including a portion on stationary point processes. Features Presents and illustrates the fundamental correlation and spectral methods for stochastic processes and random fields Explains how the basic theory is used in special applications like detection theory and signal processing, spatial statistics, and reliability Motivates mathematical theory from a statistical model-building viewpoint Introduces a selection of special topics, including extreme value theory, filter theory, long-range dependence, and point processes Provides more than 100 exercises with hints to solutions and selected full solutions This book covers key topics such as ergodicity, crossing problems, and extremes, and opens the doors to a selection of special topics, like extreme value theory, filter theory, long-range dependence, and point processes, and includes many exercises and examples to illustrate the theory. Precise in mathematical details without being pedantic, Stationary Stochastic Processes: Theory and Applications is for the student with some experience with stochastic processes and a desire for deeper understanding without getting bogged down in abstract mathematics.

Written with students and professors in mind, Analysis of Queues: Methods and Applications combines coverage of classical queueing theory with recent advances in studying stochastic networks. Exploring a broad range of applications, the book contains plenty of solved problems, exercises, case studies, paradoxes, and numerical examples. In addition to the standard single-station and single class discrete queues, the book discusses models for multi-class queues and queueing networks as well as methods based on fluid scaling, stochastic fluid flows, continuous parameter Markov processes, and quasi-birth-and-death processes, to name a few. It describes a variety of applications including computer-communication networks, information systems, production operations, transportation, and service systems such as healthcare, call centers and restaurants.

This comprehensive guide to stochastic processes gives a complete overview of the theory and addresses the most important applications. Pitched at a level accessible to beginning graduate students and researchers from applied disciplines, it is both a course book and a rich resource for individual readers. Subjects covered include Brownian motion, stochastic calculus, stochastic differential equations, Markov processes, weak convergence of processes and semigroup theory. Applications include the Black-Scholes formula for the pricing of derivatives in financial mathematics, the Kalman-Bucy filter used in the US space program and also theoretical applications to partial differential equations and analysis. Short, readable chapters aim for clarity rather than full generality. More than 350 exercises are included to help readers put their new-found knowledge to the test and to prepare them for tackling the research literature.

The developmentsin the recent yearsof the potential theoryemphasized a classof functions larger than that of excessive functions (i.e. the positive superharmonic functionsfromtheclassicalpotentialtheoryassociatedwiththeLaplaceoperator), namely the strongly supermedian functions. It turns out that a positive Borel function will be strongly supermedian if and only if it is the in?mum of all its excessive majorants. Apparently, these functions have been introduced by J.F. Mertens and then they have been studied mainly by P.A. Meyer, G. Mokobodzki, D. Feyel and recently by P.J. Fitzsimmons and R.K. Getoor. The aimofthis bookisamongothersto developa potential theoryappropriate to this new class of functions. Although our methods are analytical, we present also the probabilistic counterparts from the Markov processes theory. The natural frame in which this theory is settled is given by a sub-Markovian resolvent of kernels on a Radon measurable space. After a possible extension of the space, such a resolvent becomes that one associated with a right process on a Radon topological space, not necessary locally compact and without existing a reference measure. Intimately related to the excessive functions we present certain basic tools of the theory: the Ray topology and compacti?cation, the ?ne carrier and the reduction operation on measurable sets. We examine di?erent types of negligible sets with respect to a ?nite measure ?: the ?-polar, ?-semipolar and ?-mince sets. We take advantage of the cone of potentials structure for both excessive functions and measures

The 3rd International ISAAC Congress took place from August 20 to 25, 2001 in Berlin, Germany, supported by the German Research Foundation (DFG), the city of Berlin through Investitionsbank Berlin and the Freie Universitiit Berlin. 10 ISAAC Awards were presented to young researchers in analysis its applications and computation from all over the world on the basis of financial support from Siemens, Daimler Crysler, Motorola and the Berlin Mathematical Society and book gifts from Birkhauser Verlag, Elsevier, Kluwer Academic Publisher, Springer Verlag and World Scientific. The ISAAC is grateful to all these institutions, firms and publishers for their support. Due to the support from DFG and from Investitions bank Berlin many of the 362 registrated participants could be financially supported. Unfortunately the financial supports were granted too late to reach more people from former SU as the procedere for visa is still more than cumbersome and embassies are not at all flexible. Hence, a big part of the financial support could not be used and had to be returned. The 10 plenary lectures were 1. Antoniou, 1. Prigogine (Intern. Solvay Inst. Phys. Chem., Brussels): Irreversibility and the probabilistic description of unstable evolutions beyond the Hilbert space framework (read by 1. Antoniou), N.S. Bakhvalov, M.E. Eglit (Math. Mech. Dept., Lomonosov State Univ."

This book is intended as a continuation of my book "Parametrix Method in the Theory of Differential Complexes" (see [291]). There, we considered complexes of differential operators between sections of vector bundles and we strived more than for details. Although there are many applications to for maximal generality overdetermined systems, such an approach left me with a certain feeling of dissat- faction, especially since a large number of interesting consequences can be obtained without a great effort. The present book is conceived as an attempt to shed some light on these new applications. We consider, as a rule, differential operators having a simple structure on open subsets of Rn. Currently, this area is not being investigated very actively, possibly because it is already very highly developed actively (cf. for example the book of Palamodov [213]). However, even in this (well studied) situation the general ideas from [291] allow us to obtain new results in the qualitative theory of differential equations and frequently in definitive form. The greater part of the material presented is related to applications of the L- rent series for a solution of a system of differential equations, which is a convenient way of writing the Green formula. The culminating application is an analog of the theorem of Vitushkin [303] for uniform and mean approximation by solutions of an elliptic system. Somewhat afield are several questions on ill-posedness, but the parametrix method enables us to obtain here a series of hitherto unknown facts.

The monograph presents some of the authors' recent and original results concerning boundedness and compactness problems in Banach function spaces both for classical operators and integral transforms defined, generally speaking, on nonhomogeneous spaces. Itfocuses onintegral operators naturally arising in boundary value problems for PDE, the spectral theory of differential operators, continuum and quantum mechanics, stochastic processes etc. The book may be considered as a systematic and detailed analysis of a large class of specific integral operators from the boundedness and compactness point of view. A characteristic feature of the monograph is that most of the statements proved here have the form of criteria. These criteria enable us, for example, togive var ious explicit examples of pairs of weighted Banach function spaces governing boundedness/compactness of a wide class of integral operators. The book has two main parts. The first part, consisting of Chapters 1-5, covers theinvestigation ofclassical operators: Hardy-type transforms, fractional integrals, potentials and maximal functions. Our main goal is to give a complete description of those Banach function spaces in which the above-mentioned operators act boundedly (com pactly). When a given operator is not bounded (compact), for example in some Lebesgue space, we look for weighted spaces where boundedness (compact ness) holds. We develop the ideas and the techniques for the derivation of appropriate conditions, in terms of weights, which are equivalent to bounded ness (compactness)."

Analytic Extension is a mysteriously beautiful property of analytic functions. With this point of view in mind the related survey papers were gathered from various fields in analysis such as integral transforms, reproducing kernels, operator inequalities, Cauchy transform, partial differential equations, inverse problems, Riemann surfaces, Euler-Maclaurin summation formulas, several complex variables, scattering theory, sampling theory, and analytic number theory, to name a few. Audience: Researchers and graduate students in complex analysis, partial differential equations, analytic number theory, operator theory and inverse problems.

In recent years approximation theory and the theory of orthogonal polynomials have witnessed a dramatic increase in the number of solutions of difficult and previously untouchable problems. This is due to the interaction of approximation theoretical techniques with classical potential theory (more precisely, the theory of logarithmic potentials, which is directly related to polynomials and to problems in the plane or on the real line). Most of the applications are based on an exten sion of classical logarithmic potential theory to the case when there is a weight (external field) present. The list of recent developments is quite impressive and includes: creation of the theory of non-classical orthogonal polynomials with re spect to exponential weights; the theory of orthogonal polynomials with respect to general measures with compact support; the theory of incomplete polynomials and their widespread generalizations, and the theory of multipoint Pade approximation. The new approach has produced long sought solutions for many problems; most notably, the Freud problems on the asymptotics of orthogonal polynomials with a respect to weights of the form exp(-Ixl ); the "l/9-th" conjecture on rational approximation of exp(x); and the problem of the exact asymptotic constant in the rational approximation of Ixl. One aim of the present book is to provide a self-contained introduction to the aforementioned "weighted" potential theory as well as to its numerous applications. As a side-product we shall also fully develop the classical theory of logarithmic potentials."