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The book incorporates research papers and surveys written byparticipants ofan International Scientific Programme onApproximation Theory jointly supervised by Institute forConstructive Mathematics of University of South Florida atTampa, USA and the Euler International MathematicalInstituteat St. Petersburg, Russia. The aim of theProgramme was to present newdevelopments in ConstructiveApproximation Theory. The topics of the papers are:asymptotic behaviour of orthogonal polynomials,rationalapproximation of classical functions, quadrature formulas,theory of n-widths, nonlinear approximation in Hardyalgebras,numerical results on best polynomialapproximations, waveletanalysis.FROM THE CONTENTS: E.A. Rakhmanov: Strong asymptotics fororthogonal polynomials associated with exponential weightson R.- A.L. Levin, E.B. Saff: Exact Convergence Rates forBest Lp Rational Approximation to the Signum Function andfor Optimal Quadrature in Hp.- H.Stahl: Uniform RationalApproximation of x .- M. Rahman, S.K. Suslov: ClassicalBiorthogonal Rational Functions.- V.P. Havin, A. PresaSague: Approximation properties of harmonic vector fieldsand differential forms.- O.G. Parfenov: Extremal problemsfor Blaschke products and N-widths.- A.J.Carpenter, R.S.Varga: Some Numerical Results on Best Uniform PolynomialApproximation of x on 0,1 .- J.S. Geronimo:PolynomialsOrthogonal on the Unit Circle with Random RecurrenceCoefficients.- S. Khrushchev: Parameters oforthogonalpolynomials.- V.N. Temlyakov: The universality oftheFibonacci cubature formulas. ... Read more

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The fundamental question of characterizing continuity and boundedness of Gaussian processes goes back to Kolmogorov. After contributions by R. Dudley and X. Fernique, it was solved by the author. This book provides an overview of "generic chaining", a completely natural variation on the ideas of Kolmogorov. It takes the reader from the first principles to the edge of current knowledge and to the open problems that remain in this domain.

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This volume focuses on the analysis of solutions to generalelliptic equations. A wide range of topics is touched upon, such asremovable singularities, Laurent expansions, approximation bysolutions, Carleman formulas, quasiconformality. While the basicsetting is the Dirichlet problem for the Laplacian, there is somediscussion of the Cauchy problem. Care is taken to distinguish betweenresults which hold in a very general setting (arbitrary ellipticequation with the unique continuation property) and those which holdunder more restrictive assumptions on the differential operators(homogeneous, of first order). Some parallels to the theory offunctions of several complex variables are also sketched.
Audience: This book will be of use to postgraduate students andresearchers whose work involves partial differential equations,approximations and expansion, several complex variables and analyticspaces, potential theory and functional analysis. It can berecommended as a text for seminars and courses, as well as forindependent study. ... Read more

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Tauberian theory compares summability methods for series and integrals, helps to decide when there is convergence, and provides asymptotic and remainder estimates. The author shows the development of the theory from the beginning and his expert commentary evokes the excitement surrounding the early results. He shows the fascination of the difficult Hardy-Littlewood theorems and of an unexpected simple proof, and extolls Wiener's breakthrough based on Fourier theory. There are the spectacular "high-indices" theorems and Karamata's "regular variation", which permeates probability theory. The author presents Gelfand's elegant algebraic treatment of Wiener theory and his own distributional approach. There is also a new unified theory for Borel and "circle" methods. The text describes many Tauberian ways to the prime number theorem. A large bibliography and a substantial index round out the book.

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Many nonlinear problems in physics, engineering, biology and social sciences can be reduced to finding critical points of functionals. While minimax and Morse theories provide answers to many situations and problems on the existence of multiple critical points of a functional, they often cannot provide much-needed additional properties of these critical points. Sign-changing critical point theory has emerged as a new area of rich research on critical points of a differentiable functional with important applications to nonlinear elliptic PDEs.

This book is intended for advanced graduate students and researchers involved in sign-changing critical point theory, PDEs, global analysis, and nonlinear functional analysis.

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The main features of this volume are: 1) It is devoted to the basic principles of asymptotics and their applications; 2) It presents both traditional approaches as well as less widely used and new approaches such as one- and two-point Padé Approximants, constitutive equations, methods of boundary perturbations, etc.; 3) A general introduction to the subject suitable for non-specialists. Compared with other published books in the field the authors have paid special attention to examples and the discussion of results rather than burying them in formalism, in notation and in technical details. Audience: Researchers in mechanics, physics and applied mathematics as well as in engineering. Graduate students and even high school students can benefit from reading the book, which does not require any scientific knowledge of mathematics and physics. ... Read more

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This volume aims to disseminate a number of new ideas that have emerged in the last few years in the field of numerical simulation, all bearing the common denominator of the ... Read more

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The method of exponential sums is a general method enabling thesolution of a wide range of problems in the theory of numbers and itsapplications. This volume presents an exposition of the fundamentalsof the theory with the help of examples which show how exponentialsums arise and how they are applied in problems of number theory andits applications.
The material is divided into three chapters which embrace theclassical results of Gauss, and the methods of Weyl, Mordell andVinogradov; the traditional applications of exponential sums to thedistribution of fractional parts, the estimation of the Riemann zetafunction; and the theory of congruences and Diophantine equations.Some new applications of exponential sums are also included.
It is assumed that the reader has a knowledge of the fundamentals ofmathematical analysis and of elementary number theory.
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Customer Reviews (1)

A Great Book
This great book is written by
one of the greatest number theorists; N. M. Korobov.
In this book is very nicely and lucid explained the most powerful tool
in analytic number theory; "Exponential Sums". ... Read more

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Accessible to anyone with a good general background in mathematics, but it nonetheless gets right to the cutting edge of active research.

Some results appear here for the first time, while others have hitherto only been given in preprints.

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This book contains the proceedings of the International Workshop on Operator Theory and Applications held in Faro, Portugal, September 12 to 15, 2000. It includes 20 selected articles centered on the analysis of various classes of singular operators, the factorization of operator and matrix functions, algebraic methods in approximation theory, and applications in diffraction theory. Some papers are related to topics from fractional calculus, complex analysis, operator algebras, and partial differential equations.

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In 1909 Alfred Haar introduced into analysis a remarkablesystem which bears his name. The Haar system is a complete orthonormalsystem on [0,1] and the Fourier-Haar series for arbitrarycontinuous function converges uniformly to this function.
This volume is devoted to the investigation of the Haar system fromthe operator theory point of view. The main subjects treated are:classical results on unconditional convergence of the Haar series inmodern presentation; Fourier-Haar coefficients;reproducibility; martingales; monotone bases in rearrangementinvariant spaces; rearrangements and multipliers with respect to theHaar system; subspaces generated by subsequences of the Haar system;the criterion of equivalence of the Haar and Franklin systems.
Audience: This book will be of interest to graduate students andresearchers whose work involves functional analysis and operatortheory. ... Read more

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The book establishes links between regularity and derivativeconcepts of nonsmooth analysis and studies of solution methods andstability for optimization, complementarity and equilibrium problems.In developing necessary tools, it presents, in particular:

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This volume provides a comprehensive introduction to the theoryof spline functions. Emphasis is given to new developments, such asthe general Birkhoff-type interpolation, the extremal properties ofsplines, their prominent role in the optimal recovery of functions,and multivariate interpolation by polynomials and splines.
The book has thirteen chapters dealing, respectively, withinterpolation by algebraic polynomials, the space of splines,B-splines, interpolation by spline functions, natural splinefunctions, perfect splines, monosplines, periodic splines,multivariate B-splines and truncated powers, multivariate splinefunctions and divided differences, box splines, multivariate meanvalue interpolation, multivariate polynomial interpolations arising byhyperplanes, and multivariate pointwise interpolation.
Some of the results described are presented as exercises and hints aregiven for their solution.
For researchers and graduate students whose work involvesapproximation theory. ... Read more

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This volume contains articles that document the advances in thesubject of Total Positivity during the last two decades. The materialis divided into ten chapters. While some of the articles are of asurvey nature, others present new results appearing here for the firsttime. Also, some papers contain introductory material and aretherefore accessible to non-experts interested in becoming familiarwith the important ideas and techniques of Total Positivity.
Audience: This book will be of value to mathematicians,engineers and computer scientists whose work involves applications ofTotal Positivity to problems in the theory of spline functions,numerical quadrature, nonlinear analysis, entire functions,probability, mathematical biology, statistics, approximation theory,combinatorics, geometric modelling, matrix theory and integralequations. ... Read more

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This application-oriented monograph presents a comprehensivetheoretical and numerical investigation of all types of oscillatorsand bifurcations (such as Hopf, Bogdanov-Takens, Bautin, andhomoclinic) generated by the FitzHugh-Nagumo model. The widediversity of the oscillators as used in electrophysiology, biology,and engineering is emphasised. Various asymptotic behaviours arerevealed. The dramatic changes in oscillations connected with theemergence or disappearance of concave limit cycles are investigated.Codimension of bifurcations is minutely analysed. New types ofcodimension one and two bifurcations of planar systems were found. Adetailed global bifurcation diagram concludes the work.
Audience: This volume will be of interest to researchers andgraduate students whose work involves the mathematics of biology,ordinary differential equations, approximations and expansions,cardiac electrophysiology, biological transport, and cell membranes. ... Read more

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This book treats several topics on the global analysis oflinear differential equations in the complex domain, using originalmethods of analysis developed by K. Okubo and the present author. Forexample, global analysis of linear differential equations is closelyrelated to that of linear difference equations. Through the asymptoticbehaviour of solutions of linear difference equations, connectionproblems for hypergeometric systems can be solved, i.e. Fuchsiandifferential equations and Stokes phenomenon for Birkhoff systems.Furthermore, the calculation of monodromy groups is explained indetail. Audience: This volume will be of interest to researchers andgraduate students whose work involves ordinary differential equations,special functions, finite difference, functional equations,approximations and expansions, and symbolic/algebraic manipulation. ... Read more

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This volume considers various methods for constructing cubatureand quadrature formulas of arbitrary degree. These formulas areintended to approximate the calculation of multiple and conventionalintegrals over a bounded domain of integration. The latter is assumedto have a piecewise-smooth boundary and to be arbitrary in otheraspects. Particular emphasis is placed on invariant cubature formulasand those for a cube, a simplex, and other polyhedra. Here, thetechniques of functional analysis and partial differential equationsare applied to the classical problem of numerical integration, toestablish many important and deep analytical properties of cubatureformulas. The prerequisites of the theory of many-dimensional discretefunction spaces and the theory of finite differences are conciselypresented. Special attention is paid to constructing and studying theoptimal cubature formulas in Sobolev spaces. As an asymptoticallyoptimal sequence of cubature formulas, a many-dimensional abstractionof the Gregory quadrature is indicated.
Audience: This book is intended for researchers having a basicknowledge of functional analysis who are interested in theapplications of modern theoretical methods to numerical mathematics. ... Read more

Customer Reviews (1)

excellent book
Excellent, very useful mathematical boo ... Read more