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The manuscript gives a coherent and detailed account of the theory of series in the eighteenth and early nineteenth centuries. It provides in one place an account of many results that are generally to be found - if at all - scattered throughout the historical and textbook literature. It presents the subject from the viewpoint of the mathematicians of the period, and is careful to distinguish earlier conceptions from ones that prevail today.

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First monograph which considers the theory of more parameter dyadic and classical Hardy spaces. Useful for researchers as well as graduate or postgraduate students whose work involved Fourier analysis, probability theory, stochastic processes and complex variables. ... Read more

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The main topic of this book is the theory of bifurcations ofvector fields, i.e. the study of families of vector fields dependingon one or several parameters and the changes (bifurcations) in thetopological character of the objects studied as parameters vary. Inparticular, one of the phenomena studied is the bifurcation ofperiodic orbits from a singular point or a polycycle. The followingtopics are discussed in the book:

• Divergentseries and resummation techniques with applications, in particular tothe proofs of the finiteness conjecture of Dulac saying thatpolynomial vector fields on R² cannot possess an infinity oflimit cycles. The proofs work in the more general context of realanalytic vector fields on the plane.
• Techniques in the study ofunfoldings of singularities of vector fields (blowing up, normalforms, desingularization of vector fields). Local dynamics andnonlocal bifurcations.
• Knots and orbit genealogies inthree-dimensional flows.
• Bifurcations and applications:computational studies of vector fields.
• Holomorphic differentialequations in dimension two.
• Studies of real and complexpolynomial systems and of the complex foliations arising frompolynomial differential equations.
• Applications of computeralgebra to dynamical systems.

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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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This book offers basic theory on hypercomplex-analytic automorphic forms and functions for arithmetic subgroups of the Vahlen group in higher dimensional spaces. Vector- and Clifford algebra-valued Eisenstein and Poincaré series are constructed within this framework and a detailed description of their analytic and number theoretical properties is provided. It establishes explicit relationships to generalized variants of the Riemann zeta function and Dirichlet L-series, and introduces hypercomplex multiplication of lattices.

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The book treats four mathematical concepts which play a great role in many different areas of mathematics: symbolic sums, recurrence (difference) equations, generating functions, and asymptotic estimates. Their key features in isolation or in combination, their mastery by paper and pencil or by computer programs, and their applications to problems in pure mathematics or to "real world problems" are studied. The book can be viewed as an algorithmic version of the bestselling "Concrete Mathematics" by Graham, Knuth and Patashnik. ... Read more

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The theory of functional equations has been developed in arapid and productive way in the second half of the Twentieth Century.This is due to the fact that the mathematical applications increasedthe number of investigations of newer and newer types of functionalequations. At the same time, the self-development of this theory wasalso very fruitful. The material of this volume reflects very well the complexity andapplicability of the most active research fields. The results andmethods contained give a representative crossection of what isrecently happening in the theory of functional equations. Audience: Researchers and advanced students. ... Read more

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A Concise Approach to Mathematical Analysis introduces the undergraduate student to the more abstract concepts of advanced calculus. The main aim of the book is to smooth the transition from the problem-solving approach of standard calculus to the more rigorous approach of proof-writing and a deeper understanding of mathematical analysis. The first half of the textbook deals with the basic foundation of analysis on the real line; the second half introduces more abstract notions in mathematical analysis. Each topic begins with a brief introduction followed by detailed examples. A selection of exercises, ranging from the routine to the more challenging, then gives students the opportunity to practise writing proofs. The book is designed to be accessible to students with appropriate backgrounds from standard calculus courses but with limited or no previous experience in rigorous proofs. It is written primarily for advanced students of mathematics - in the 3rd or 4th year of their degree - who wish to specialise in pure and applied mathematics, but it will also prove useful to students of physics, engineering and computer science who also use advanced mathematical techniques. ... Read more

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This book is devoted to the asymptotic theory of differentialequations. Asymptotic theory is an independent and important branch ofmathematical analysis that began to develop at the end of the 19thcentury. Asymptotic methods' use of several important phenomena ofnature can be explained.
The main problems considered in the text are based on the notion of anasymptotic space, which was introduced by the author in his works.Asymptotic spaces for asymptotic theory play analogous roles as metricspaces for functional analysis. It allows one to consider many(seemingly) miscellaneous asymptotic problems by means of the samemethods and in a compact general form.
The book contains the theoretical material and general methods of itsapplication to many partial problems, as well as several new resultsof asymptotic behavior of functions, integrals, and solutions ofdifferential and difference equations.
Audience: The material will be of interest to mathematicians,researchers, and graduate students in the fields of ordinarydifferential equations, finite differences and functional equations,operator theory, and functional analysis. ... Read more

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This volume contains a collection of papers in Analytic andElementary Number Theory in memory of Professor Paul Erdös, oneof the greatest mathematicians of this century. Written by manyleading researchers, the papers deal with the most recent advances ina wide variety of topics, including arithmetical functions, primenumbers, the Riemann zeta function, probabilistic number theory,properties of integer sequences, modular forms, partitions, andq-series.
Audience: Researchers and students of number theory, analysis,combinatorics and modular forms will find this volume to bestimulating. ... Read more

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This monograph proposes a new classification of periodicfunctions, based on the concept of generalized derivative, defined byintroducing multiplicators and shifts of the argument into the Fourierseries of the original function. This approach permits theclassification of a wide range of functions, including those of whichthe Fourier series may diverge in integral metric, smooth functions,and infinitely differentiable functions, including analytical andentire ones. These newly introduced classes are then investigatedusing the traditional problems of the theory of approximation. Theresults thus obtained offer a new way to look at classical statementsfor the approximation of differentiable functions, and suggestpossibilities to discover new effects. Audience: valuable reading for experts in the field ofmathematical analysis and researchers and graduate students interestedin the applications of the theory of approximation and Fourier series. ... Read more

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In Fourier Analysis and Approximation of Functions basics of classical Fourier Analysis are given as well as those of approximation by polynomials, splines and entire functions of exponential type.
In Chapter 1 which has an introductory nature, theorems on convergence, in that or another sense, of integral operators are given. In Chapter 2 basic properties of simple and multiple Fourier series are discussed, while in Chapter 3 those of Fourier integrals are studied.
The first three chapters as well as partially Chapter 4 and classical Wiener, Bochner, Bernstein, Khintchin, and Beurling theorems in Chapter 6 might be interesting and available to all familiar with fundamentals of integration theory and elements of Complex Analysis and Operator Theory. Applied mathematicians interested in harmonic analysis and/or numerical methods based on ideas of Approximation Theory are among them.
In Chapters 6-11 very recent results are sometimes given in certain directions. Many of these results have never appeared as a book or certain consistent part of a book and can be found only in periodics; looking for them in numerous journals might be quite onerous, thus this book may work as a reference source.
The methods used in the book are those of classical analysis, Fourier Analysis in finite-dimensional Euclidean space Diophantine Analysis, and random choice. ... Read more

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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.

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This is a revision of an earlier Means and Their Inequalities by the present author and Professors Mitrinovic and Vasic. Not only does this book bring the earlier version up to date but enlarges the scope considerably to give a full and in-depth treatment of all aspects of the field. While the mention of means occurs in many books this is the only full treatment of the subject. Outstanding features of the book are the variety of proofs given for many of the basic results, over seventy for the inequality between the arithmetic and geometric means for instance, an exhaustive bibliography and a list of mathematicians who have contributed to this field from the time of Euclid to the present day.

Audience: This book is written in a language that not only the expert on the subject will understand and appreciate, but graduate students worldwide as well. Any person with an interest in means and their inequalities should find this book within their comprehension although to fully appreciate all the topics covered a knowledge of calculus and of elementary real analysis is required.

Customer Reviews (1)

Accessible for the serious professional
This book is just what I was looking for ... and so much more. Here is a complete and in-depth treatment of the subject of means. Chapter 2 ... on arithmetic, geometric, and harmonic means ... is the primary reason I purchased the book ... and I'm NOT disappointed. ... Read more

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The classical books on interpolation address numerous negative results, i.e., results on divergent interpolation processes, usually constructed over some equidistant systems of nodes. The authors present, with complete proofs, recent results on convergent interpolation processes, for trigonometric and algebraic polynomials of one real variable, not yet published in other textbooks and monographs on approximation theory and numerical mathematics. In this special, but fundamental and important field of real analysis the authors present the state of art. Some 500 references are cited, including many new results of the authors. Basic tools in this field (orthogonal polynomials, moduli of smoothness, K-functionals, etc.) as well as some selected applications in numerical integration, integral equations, moment-preserving approximation and summation of slowly convergent series are also given. Beside the basic properties of the classical orthogonal polynomials the book provides new results on nonclassical orthogonal polynomials including methods for their numerical construction.

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The theory of functional equations has been developed in a rapid and productive way in the second half of the Twentieth Century. This is due to the fact that the mathematical applications increased the number of investigations of newer and newer types of functional equations. At the same time, the self-development of this theory was also very fruitful.
The material of this volume reflects very well the complexity and applicability of the most active research fields. The results and methods contained give a representative crossection of what is recently happening in the theory of functional equations. 
... Read more

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This is the first monograph which considers the theory of more-parameter dyadic and classical Hardy spaces. In this book a new application of martingale and distribution theories is dealt with. The theories of the multi-parameter dyadic martingale and the classical Hardy spaces are applied in Fourier analysis. Several summability methods of d-dimensional trigonometric-, Walsh-, spline-, and Ciesielski-Fourier series and Fourier transforms as well as the d-dimensional dyadic derivative are investigated. The boundedness of the maximal operators of the summations on Hardy spaces, weak (L₁, L₁) inequalities and a.e. convergence results for the d-dimensional Fourier series are proved.

Audience: This book will be useful for researchers as well as for graduate or postgraduate students whose work involves Fourier analysis, approximations and expansions, sequences, series, summability, probability theory, stochastic processes, several complex variables, and analytic spaces.

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This text is a rigorous, detailed introduction to real analysis that presents the fundamentals with clear exposition and carefully written definitions, theorems, and proofs. It is organized in a distinctive, flexible way that would make it equally appropriate to undergraduate mathematics majors who want to continue in mathematics, and to future mathematics teachers who want to understand the theory behind calculus.

Key Features:

Provides an unusually thorough treatment of the real numbers, emphasizing their importance as the basis of real analysis,

Presents material in an order resembling that of standard calculus courses, for the sake of student familiarity, and for helping future teachers use real analysis to better understand calculus,

Emphasizes the direct role of the Least Upper Bound Property in the study of limits, derivatives and integrals, rather than making use of sequences for proofs,

Presents the equivalence of various important theorems of real analysis with the Least Upper Bound Property,

Relates real analysis to previously learned material, including detailed discussion of such topics as the transcendental functions, area and the number pi,

Offers three different entryways into the study of real numbers, depending on the student audience,

Contains historical context, biographical anecdotes, and reflections on the material in each chapter.

The text is accessible to students with prior experience with mathematical proofs who have not previously studied real analysis. The text includes over 350 exercises.

The Real Numbers and Real Analysis will serve as an excellent one-semester text for undergraduates majoring in mathematics, and for students in mathematics education who want a thorough understanding of the theory behind the real number system and calculus.

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This is the first book dedicated to covering the basic elementsof the Gibbs phenomenon as it appears in various applications wherefunctions with jump discontinuities are represented. It is presentedwith detailed analysis and illustrations combined with historicalinformation. The author covers the appearance of the Gibbs phenomenonin Fourier analysis, orthogonal expansions, integral transforms,splines and wavelet approximations. Methods of reducing, or filteringout, such phenomena that cover all the above function representationsare also addressed. The book includes a thorough bibliography of some350 references.
Audience: The work is intended as an introduction forengineering and scientific practitioners in the fields where thisphenomenon may appear in their use of various functionrepresentations. It may also be used by qualified students. ... Read more