Historically, complex analysis and geometrical function theory have been inten sively developed from the beginning of the twentieth century. They provide the foundations for broad areas of mathematics. In the last fifty years the theory of holomorphic mappings on complex spaces has been studied by many mathemati cians with many applications to nonlinear analysis, functional analysis, differential equations, classical and quantum mechanics. The laws of dynamics are usually presented as equations of motion which are written in the abstract form of a dy namical system: dx / dt + f ( x) = 0, where x is a variable describing the state of the system under study, and f is a vector function of x. The study of such systems when f is a monotone or an accretive (generally nonlinear) operator on the under lying space has been recently the subject of much research by analysts working on quite a variety of interesting topics, including boundary value problems, integral equations and evolution problems (see, for example, [19, 13] and [29]). In a parallel development (and even earlier) the generation theory of one parameter semigroups of holomorphic mappings in en has been the topic of interest in the theory of Markov stochastic processes and, in particular, in the theory of branching processes (see, for example, [63, 127, 48] and [69]).

This book is a collection of the various old and new results, centered around the following simple and beautiful observation of J.L. Walsh - If a function is analytic in a finite disc, and not in a larger disc, then the difference between the Lagrange interpolant of the function, at the roots of unity, and the partial sums of the Taylor series, about the origin, tends to zero in a larger disc than the radius of convergence of the Taylor series, while each of these operators converges only in the original disc.

This book will be particularly useful for researchers in approximation and interpolation theory.

In the three decades since the introduction of the Kobayashi distance, the subject of hyperbolic complex spaces and holomorphic mappings has grown to be a big industry. This book gives a comprehensive and systematic account on the Carath odory and Kobayashi distances, hyperbolic complex spaces and holomorphic mappings with geometric methods. A very complete list of references should be useful for prospective researchers in this area.

Onc service malhemalics has rendered Ihe "Et moil ... si ravait au oomment en revcnir. je n'y serais point aU' ' human race. It has put common sense back whcre it belongs, on the topmost shelf next Iules Verne to the dUlty canister IabeUed 'discarded n- sense'. The series is divergent; therefore we may be Eric T. BeU able to do something with it. O. H eaviside Mathematics is a tool for thought, A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other pans and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'are of this series."

Number theory has a wealth of long-standing problems, the study of which over the years has led to major developments in many areas of mathematics. This volume consists of seven significant chapters on number theory and related topics. Written by distinguished mathematicians, key topics focus on multipartitions, congruences and identities (G. Andrews), the formulas of Koshliakov and Guinand in Ramanujan's Lost Notebook (B. C. Berndt, Y. Lee, and J. Sohn), alternating sign matrices and the Weyl character formulas (D. M. Bressoud), theta functions in complex analysis (H. M. Farkas), representation functions in additive number theory (M. B. Nathanson), and mock theta functions, ranks, and Maass forms (K. Ono), and elliptic functions (M. Waldschmidt).

The First International Congress of the International Society for Analysis, its Applications and Computations (ISAAC'97) was held at the University of Delaware from 3 to 7 June 1997. As specified in the invitation of the President Professor Robert P. Gilbert of the ISAAC, we organized the session on Reproducing Kerneis and Their Applications. In our session, we presented 24 engaging talks on topics of current interest to the research community. As suggested and organized by Professor Gilbert, we hereby publish its Proceedings. Rather than restricting the papers to Congress participants, we asked the Ieading mathematicians in the field of the theory of reproducing kern eIs to submit papers. However, due to time restrietions and a compulsion to limit the Proceedings a reasonable size, we were unable to obtain a comprehensive treatment of the theory of reproducing kernels. Nevertheless, we hope this Proceedings of the First International Conference on reproducing kerneis will become a significant reference volume. Indeed, we believe that the theory of reproducing kernels will stand out as a fundamental and beautiful contribution in mathematical sciences with a broad array of applications to other areas of mathematics and science. We would like to thank Professor Robert Gilbert for his substantial contri bu tions to the Congress and to our Proceedings. We also express our sincere thanks to the staff of the University of Delaware for their manifold cooperation in organizing the Congress."

This book deals with the theory of Kac algebras and their dual ity, elaborated independently by M. Enock and J . -M. Schwartz, and by G. !. Kac and L. !. Vajnermann in the seventies. The sub ject has now reached a state of maturity which fully justifies the publication of this book. Also, in recent times, the topic of "quantum groups" has become very fashionable and attracted the attention of more and more mathematicians and theoret ical physicists. One is still missing a good characterization of quantum groups among Hopf algebras, similar to the character ization of Lie groups among locally compact groups. It is thus extremely valuable to develop the general theory, as this book does, with emphasis on the analytical aspects of the subject instead of the purely algebraic ones. The original motivation of M. Enock and J. -M. Schwartz can be formulated as follows: while in the Pontrjagin duality theory of locally compact abelian groups a perfect symmetry exists between a group and its dual, this is no longer true in the various duality theorems of T. Tannaka, M. G. Krein, W. F. Stinespring . . . dealing with non abelian locally compact groups. The aim is then, in the line proposed by G. !. Kac in 1961 and M. Takesaki in 1972, to find a good category of Hopf algebras, containing the category of locally compact groups and fulfilling a perfect duality.

This book explores the theory of abelian varieties over the field of complex numbers, explaining both classic and recent results in modern language. The second edition adds five chapters on recent results including automorphisms and vector bundles on abelian varieties, algebraic cycles and the Hodge conjecture. ." . . far more readable than most . . . it is also much more complete." Olivier Debarre in Mathematical Reviews, 1994.

In the Riemann zeta function ?(s), the non-real zeros or Riemann zeros, denoted ?, play an essential role mainly in number theory, and thereby g- erate considerable interest. However, they are very elusive objects. Thus, no individual zero has an analytically known location; and the Riemann - pothesis, which states that all those zeros should lie on the critical line, i.e., 1 haverealpart, haschallengedmathematicianssince1859(exactly150years 2 ago). For analogous symmetric sets of numbers{v}, such as the roots of a k polynomial, the eigenvalues of a ?nite or in?nite matrix, etc., it is well known that symmetric functions of the{v} tend to have more accessible properties k than the individual elements v . And, we ?nd the largest wealth of explicit k properties to occur in the (generalized) zeta functions of the generic form 's Zeta(s, a)= (v ]a) k k (with the extra option of replacing v here by selected functions f(v )). k k Not surprisingly, then, zeta functions over the Riemann zeros have been considered, some as early as 1917.What is surprising is how small the lite- ture on those zeta functions has remained overall.We were able to spot them in barely a dozen research articles over the whole twentieth century and in none ofthebooks featuring the Riemannzeta function. So the domainexists, but it has remained largely con?dential and sporadically covered, in spite of a recent surge of interest. Could it then be that those zeta functions have few or uninteresting pr- erties?Inactualfact, theirstudyyieldsanabundanceofquiteexplicitresu

In its first six chapters this 2006 text seeks to present the basic ideas and properties of the Jacobi elliptic functions as an historical essay, an attempt to answer the fascinating question: 'what would the treatment of elliptic functions have been like if Abel had developed the ideas, rather than Jacobi?' Accordingly, it is based on the idea of inverting integrals which arise in the theory of differential equations and, in particular, the differential equation that describes the motion of a simple pendulum. The later chapters present a more conventional approach to the Weierstrass functions and to elliptic integrals, and then the reader is introduced to the richly varied applications of the elliptic and related functions. Applications spanning arithmetic (solution of the general quintic, the functional equation of the Riemann zeta function), dynamics (orbits, Euler's equations, Green's functions), and also probability and statistics, are discussed.

In its first six chapters this 2006 text seeks to present the basic ideas and properties of the Jacobi elliptic functions as an historical essay, an attempt to answer the fascinating question: 'what would the treatment of elliptic functions have been like if Abel had developed the ideas, rather than Jacobi?' Accordingly, it is based on the idea of inverting integrals which arise in the theory of differential equations and, in particular, the differential equation that describes the motion of a simple pendulum. The later chapters present a more conventional approach to the Weierstrass functions and to elliptic integrals, and then the reader is introduced to the richly varied applications of the elliptic and related functions. Applications spanning arithmetic (solution of the general quintic, the functional equation of the Riemann zeta function), dynamics (orbits, Euler's equations, Green's functions), and also probability and statistics, are discussed.

A. Andreotti: Nine lectures on complex analysis.- J.J. Kohn: Propagations of singularities for the Cauchy-Riemann equations.- Yum-Tong Siu: The mixed case of the direct image theorem and its applications.

One service mathematics has rendered the "Et moi, ..., si j'avait su comment en revenir, human race. It has put common sense back je n 'y serais point all

The asymptotic theory deals with the problern of determining the behaviour of a function in a neighborhood of its singular point. The function is replaced by another known function ( named the asymptotic function) close (in a sense) to the function under consideration. Many problems of mathematics, physics, and other divisions of natural sci ence bring out the necessity of solving such problems. At the present time asymptotic theory has become an important and independent branch of mathematical analysis. The present consideration is mainly based on the theory of asymp totic spaces. Each asymptotic space is a collection of asymptotics united by an associated real function which determines their growth near the given point and (perhaps) some other analytic properties. The main contents of this book is the asymptotic theory of ordinary linear differential equations with variable coefficients. The equations with power order growth coefficients are considered in detail. As the application of the theory of differential asymptotic fields, we also consider the following asymptotic problems: the behaviour of explicit and implicit functions, improper integrals, integrals dependent on a large parameter, linear differential and difference equations, etc .. The obtained results have an independent meaning. The reader is assumed to be familiar with a comprehensive course of the mathematical analysis studied, for instance at mathematical departments of universities. Further necessary information is given in this book in summarized form with proofs of the main aspects."

An arrangement of hyperplanes is a finite collection of codimension one affine subspaces in a finite dimensional vector space. Arrangements have emerged independently as important objects in various fields of mathematics such as combinatorics, braids, configuration spaces, representation theory, reflection groups, singularity theory, and in computer science and physics. This book is the first comprehensive study of the subject. It treats arrangements with methods from combinatorics, algebra, algebraic geometry, topology, and group actions. It emphasizes general techniques which illuminate the connections among the different aspects of the subject. Its main purpose is to lay the foundations of the theory. Consequently, it is essentially self-contained and proofs are provided. Nevertheless, there are several new results here. In particular, many theorems that were previously known only for central arrangements are proved here for the first time in completegenerality. The text provides the advanced graduate student entry into a vital and active area of research. The working mathematician will findthe book useful as a source of basic results of the theory, open problems, and a comprehensive bibliography of the subject.

Nevanlinna theory (or value distribution theory) in complex analysis is so beautiful that one would naturally be interested in determining how such a theory would look in the non Archimedean analysis and Diophantine approximations. There are two "main theorems" and defect relations that occupy a central place in N evanlinna theory. They generate a lot of applications in studying uniqueness of meromorphic functions, global solutions of differential equations, dynamics, and so on. In this book, we will introduce non-Archimedean analogues of Nevanlinna theory and its applications. In value distribution theory, the main problem is that given a holomorphic curve f : C -+ M into a projective variety M of dimension n and a family 01 of hypersurfaces on M, under a proper condition of non-degeneracy on f, find the defect relation. If 01 n is a family of hyperplanes on M = r in general position and if the smallest dimension of linear subspaces containing the image f(C) is k, Cartan conjectured that the bound of defect relation is 2n - k + 1. Generally, if 01 is a family of admissible or normal crossings hypersurfaces, there are respectively Shiffman's conjecture and Griffiths-Lang's conjecture. Here we list the process of this problem: A. Complex analysis: (i) Constant targets: R. Nevanlinna[98] for n = k = 1; H. Cartan [20] for n = k > 1; E. I. Nochka [99], [100],[101] for n > k ~ 1; Shiffman's conjecture partially solved by Hu-Yang [71J; Griffiths-Lang's conjecture (open).

This book offers an elementary and engaging introduction to operator theory on the Hardy-Hilbert space. It provides a firm foundation for the study of all spaces of analytic functions and of the operators on them. Blending techniques from "soft" and "hard" analysis, the book contains clear and beautiful proofs. There are numerous exercises at the end of each chapter, along with a brief guide for further study which includes references to applications to topics in engineering.

The primary goal of this text is to present the theoretical foundation of the field of Fourier analysis. This book is mainly addressed to graduate students in mathematics and is designed to serve for a three-course sequence on the subject. The only prerequisite for understanding the text is satisfactory completion of a course in measure theory, Lebesgue integration, and complex variables. This book is intended to present the selected topics in some depth and stimulate further study. Although the emphasis falls on real variable methods in Euclidean spaces, a chapter is devoted to the fundamentals of analysis on the torus. This material is included for historical reasons, as the genesis of Fourier analysis can be found in trigonometric expansions of periodic functions in several variables.

While the 1st edition was published as a single volume, the new edition will contain 120 pp of new material, with an additional chapter on time-frequency analysis and other modern topics. As a result, the book is now being published in 2 separate volumes, the first volume containing the classical topics (Lp Spaces, Littlewood-Paley Theory, Smoothness, etc...), the second volume containing the modern topics (weighted inequalities, wavelets, atomic decomposition, etc...).

From a review of the first edition:

"Grafakos's book is very user-friendly with numerous examples illustrating the definitions and ideas. It is more suitable for readers who want to get a feel for current research. The treatment is thoroughly modern with free use of operators and functional analysis. Morever, unlike many authors, Grafakos has clearly spent a great deal of time preparing the exercises." - Ken Ross, MAA Online

Can be used as a graduate text

Contains many exercises

Contains new results

A quadratic differential on aRiemann surface is locally represented by a ho lomorphic function element wh ich transforms like the square of a derivative under a conformal change of the parameter. More generally, one also allows for meromorphic function elements; however, in many considerations it is con venient to puncture the surface at the poles of the differential. One is then back at the holomorphic case. A quadratic differential defines, in a natural way, a field of line elements on the surface, with singularities at the critical points, i.e. the zeros and poles of the differential. The integral curves of this field are called the trajectories of the differential. A large part of this book is about the trajectory structure of quadratic differentials. There are of course local and global aspects to this structure. Be sides, there is the behaviour of an individual trajectory and the structure deter mined by entire subfamilies of trajectories. An Abelian or first order differential has an integral or primitive function is in general not single-valued. In the case of a quadratic on the surface, which differential, one first has to take the square root and then integrate. The local integrals are only determined up to their sign and arbitrary additive constants. However, it is this multivalued function which plays an important role in the theory; the trajectories are the images of the horizontals by single valued branches of its inverse."

This book is an account of the theory of Hardy spaces in one dimension, with emphasis on some of the exciting developments of the past two decades or so. The last seven of the ten chapters are devoted in the main to these recent developments. The motif of the theory of Hardy spaces is the interplay between real, complex, and abstract analysis. While paying proper attention to each of the three aspects, the author has underscored the effectiveness of the methods coming from real analysis, many of them developed as part of a program to extend the theory to Euclidean spaces, where the complex methods are not available.

S.G. Gindikin, I.I. Pjateckii-Sapiro, E.B. Vinberg: Homogeneous K hler manifolds.- S.G. Greenfield: Extendibility properties of real submanifolds of Cn.- W. Kaup: Holomorphische Abbildungen in Hyperbolische R ume.- A. Koranyi: Holomorphic and harmonic functions on bounded symmetric domains.- J.L. Koszul: Formes harmoniques vectorielles sur les espaces localement sym triques.- S. Murakami: Plongements holomorphes de domaines sym triques.- E.M. Stein: The analogues of Fatous 's theorem and estimates for maximal functions.

A. Dynin: Pseudo-differential operators on Heisenberg groups.- A. Dynin: An index formula for elliptic boundary problems.- G.I. Eskin: General mixed boundary problems for elliptic differential equations.- B. Helffer: Hypoellipticite pour des operateurs differentiels sur des groupes de Lie nilpotents.- J.J. Kohn: Lectures on degenerate elliptic problems.- K. Taira: Conditions necessaires et suffisantes pour l'existence et l'unicite des solutions du probleme de la derivee oblique.- F. Treves: Boundary value problems for elliptic equations.

All modem introductions to complex analysis follow, more or less explicitly, the pattern laid down in Whittaker and Watson 75]. In "part I'' we find the foundational material, the basic definitions and theorems. In "part II" we find the examples and applications. Slowly we begin to understand why we read part I. Historically this is an anachronism. Pedagogically it is a disaster. Part II in fact predates part I, so clearly it can be taught first. Why should the student have to wade through hundreds of pages before finding out what the subject is good for? In teaching complex analysis this way, we risk more than just boredom. Beginning with a series of unmotivated definitions gives a misleading impression of complex analy sis in particular and of mathematics in general. The classical theory of analytic functions did not arise from the idle speculation of bored mathematicians on the possible conse quences of an arbitrary set of definitions; it was the natural, even inevitable, consequence of the practical need to answer questions about specific examples. In standard texts, after hundreds of pages of theorems about generic analytic functions with only the rational and trigonometric functions as examples, students inevitably begin to believe that the purpose of complex analysis is to produce more such theorems. We require introductory com plex analysis courses of our undergraduates and graduates because it is useful both within mathematics and beyond."

In the last decade, convolution operators of matrix functions have received unusual attention due to their diverse applications. This monograph presents some new developments in the spectral theory of these operators. The setting is the Lp spaces of matrix-valued functions on locally compact groups. The focus is on the spectra and eigenspaces of convolution operators on these spaces, defined by matrix-valued measures. Among various spectral results, the L2-spectrum of such an operator is completely determined and as an application, the spectrum of a discrete Laplacian on a homogeneous graph is computed using this result. The contractivity properties of matrix convolution semigroups are studied and applications to harmonic functions on Lie groups and Riemannian symmetric spaces are discussed. An interesting feature is the presence of Jordan algebraic structures in matrix-harmonic functions.