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Asymptotic methods provide important tools for approximating and analysing functions that arise in probability and statistics. Moreover, the conclusions of asymptotic analysis often supplement the conclusions obtained by numerical methods. Providing a broad toolkit of analytical methods, Expansions and Asymptotics for Statistics shows how asymptotics, when coupled with numerical methods, becomes a powerful way to acquire a deeper understanding of the techniques used in probability and statistics.

The book first discusses the role of expansions and asymptotics in statistics, the basic properties of power series and asymptotic series, and the study of rational approximations to functions. With a focus on asymptotic normality and asymptotic efficiency of standard estimators, it covers various applications, such as the use of the delta method for bias reduction, variance stabilisation, and the construction of normalising transformations, as well as the standard theory derived from the work of R.A. Fisher, H. Cramér, L. Le Cam, and others. The book then examines the close connection between saddle-point approximation and the Laplace method. The final chapter explores series convergence and the acceleration of that convergence.

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This book presents a thorough development of the modern theory of stochastic approximation or recursive stochastic algorithms for both constrained and unconstrained problems. This second edition is a thorough revision, although the main features and structure remain unchanged. It contains many additional applications and results as well as more detailed discussion.

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In addition to coverage of univariate interpolation and approximation, the text includes material on multivariate interpolation and multivariate numerical integration, a generalization of the Bernstein polynomials that has not previously appeared in book form, and a greater coverage of Peano kernel theory than is found in most textbooks. There are many worked examples and each section ends with a number of carefully selected problems that extend the student's understanding of the text. The author is well known for his clarity of writing and his many contributions as a researcher in approximation theory.

Customer Reviews (1)

Great reference book, well written
I would recommend this book to anyone interested in numerical analysis, applied mathematics or approximation theory. First because polynomials are often the foundation and this books gives you much of the knowledge and basic tricks needed to be in control of them. Secondly because this book is a little gem of clarity that will enlighten you to the point where you may start to get a glimps of the beauty of mathematics. It is very well written book and joy to read.

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This text introduces the concept of Polya property for holomorphic functions of two and several variables. The functions which have the Polya property are used to study the uniqueness problem for entire functions. This gives interesting results for a general class of generating functions. The functions with Polya property are used as generating kernels in formal generating relations to study the expansion problem for entire functions. A necessary and sufficient condition is obtained for an entire function to have a unique expansion in a series of various known polynomials generated from the formal generating relations. In this book it is shown that the uniqueness problem for entire functions of exponential type is equivalent to the approximation problem for analytic functions. Since functions with Polya property have very useful applications, the study of a class of linear operators which preserve the Polya property is embarked upon and two sufficient conditions for an operator to be Polya-property-preserving are given. ... 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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The aim of this volume is to make available to a large audiencerecent material in nonlinear functional analysis that has not beencovered in book format before. Here, several topics of current andgrowing interest are systematically presented, such as fixed pointtheory, best approximation, the KKM-map principle, and results relatedto optimization theory, variational inequalities and complementarityproblems. Illustrations of suitable applications are given, the linksbetween results in various fields of research are highlighted, and anup-to-date bibliography is included to assist readers in furtherstudies.
Audience: This book will be of interest to graduate students,researchers and applied mathematicians working in nonlinear functionalanalysis, operator theory, approximations and expansions, convex setsand related geometric topics and game theory. ... Read more

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Fixed point theory and best approximation
This is a good book on the study of fixed points in terms of applying kkm theory. Authors focus on the study of fixed point and related best approximation theory by using the concept of the KKM mapping. This bookcontains the most classic results on both metric fixed point andtopological fixed point results. It is a good reference for people who wantto know what kind of fixed point existrence theorems available for theirapplications in nonlinear analysis and associated subjects. ... Read more

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Approximation Theory, Wavelets and Applications drawstogether the latest developments in the subject, provides directionsfor future research, and paves the way for collaborative research. Themain topics covered include constructive multivariate approximation,theory of splines, spline wavelets, polynomial and trigonometricwavelets, interpolation theory, polynomial and rational approximation.Among the scientific applications were de-noising using wavelets,including the de-noising of speech and images, and signal and digitalimage processing. In the area of the approximation of functions themain topics include multivariate interpolation, quasi-interpolation,polynomial approximation with weights, knot removal for scattereddata, convergence theorems in Padé theory, Lyapunov theory inapproximation, Neville elimination as applied to shape preservingpresentation of curves, interpolating positive linear operators,interpolation from a convex subset of Hilbert space, and interpolationon the triangle and simplex.
Wavelet theory is growing extremely rapidly and has applications whichwill interest readers in the physical, medical, engineering and socialsciences.
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Approximation methods are vital in many challenging applications of computational science and engineering.

This is a collection of papers from world experts in a broad variety of relevant applications, including pattern recognition, machine learning, multiscale modelling of fluid flow, metrology, geometric modelling, tomography, signal and image processing.

It documents recent theoretical developments which have lead to new trends in approximation, it gives important computational aspects and multidisciplinary applications, thus making it a perfect fit for graduate students and researchers in science and engineering who wish to understand and develop numerical algorithms for the solution of their specific problems.

An important feature of the book is that it brings together modern methods from statistics, mathematical modelling and numerical simulation for the solution of relevant problems, with a wide range of inherent scales.

Contributions of industrial mathematicians, including representatives from Microsoft and Schlumberger, foster the transfer of the latest approximation methods to real-world applications.

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There has been much recent progress in approximation algorithmsfor nonconvex continuous and discrete problems, from both atheoretical and a practical perspective. In discrete (orcombinatorial) optimization many approaches have been developedrecently that link the discrete universe to the continuous universethrough geometric, analytic, and algebraic techniques. Such techniquesinclude global optimization formulations, semidefinite programming,and spectral theory. As a result new approximate algorithms have beendiscovered and many new computational approaches have been developed.Similarly, for many continuous nonconvex optimization problems, newapproximate algorithms have been developed based on semidefiniteprogramming and new randomization techniques. On the other hand,computational complexity, originating from the interactions betweencomputer science and numerical optimization, is one of the majortheories that have revolutionized the approach to solving optimizationproblems and to analyzing their intrinsic difficulty. The main focusof complexity is the study of whether existing algorithms areefficient for the solution of problems, and which problems are likelyto be tractable. The quest for developing efficient algorithms leadsalso to elegant general approaches for solving optimization problems,and reveals surprising connections among problems and their solutions.The two themes of approximation and complexity pervade this book.
Audience: Faculty, graduate students, and researchers inmathematical programming, computer sciences and engineering. ... Read more

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This volume consists of the proceedings of the NATO AdvancedResearch Workshop on Approximation by Solutions of PartialDifferential Equations, Quadrature Formulae, and Related Topics, whichwas held at Hanstholm, Denmark.
These proceedings include the main invited talks and contributedpapers given during the workshop. The aim of these lectures was topresent a selection of results of the latest research in the field. Inaddition to covering topics in approximation by solutions of partialdifferential equations and quadrature formulae, this volume is alsoconcerned with related areas, such as Gaussian quadratures, thePompelu problem, rational approximation to the Fresnel integral,boundary correspondence of univalent harmonic mappings, theapplication of the Hilbert transform in two dimensional aerodynamics,finely open sets in the limit set of a finitely generated Kleiniangroup, scattering theory, harmonic and maximal measures for rationalfunctions and the solution of the classical Dirichlet problem. Inaddition, this volume includes some problems in potential theory whichwere presented in the Problem Session at Hanstholm.
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The theory of convex optimization has been constantly developing over the past 30 years.  Most recently, many researchers have been studying more complicated classes of problems that still can be studied by means of convex analysis, so-called "anticonvex" and "convex-anticonvex" optimizaton problems.  This manuscript contains an exhaustive presentation of the duality for these classes of problems and some of its generalization in the framework of abstract convexity.  This manuscript will be of great interest for experts in this and related fields. ... 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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Although Normal Approximation and Asymptotic Expansions was first published in 1976, it has gained new significance and renewed interest among statisticians due to the developments of modern statistical techniques such as the bootstrap, the efficacy of which can be ascertained by asymptotic expansions.

This also is the only book containing a detailed treatment of various refinements of the multivariate central limit theorem (CLT), including Berry Essen-type error bounds for probabilities of general classes of functions and sets, and asymptotic expansions for both lattice and non-lattice distributions. With meticulous care, the authors develop necessary background on weak convergence theory, Fourier analysis, geometry of convex sets, and the relationship between lattice random vectors and discrete subgroups of Rk. The formalism developed in the book has been used in the extension of the theory by Goetze and Hipp to sums of weakly dependent random vectors.

This edition of the book includes a new chapter that provides an application of Stein's method of approximation to the multivariate CLT.

Audience: The book is appropriate for graduate students of probability and statistics as well as researchers in these and other fields whose work involves the asymptotic theory of statistics.

Contents: Preface to the Classics Edition; Preface; Chapter 1: Weak Convergence of Probability Measures and Uniformity Classes; Chapter 2: Fourier Transforms and Expansions of Characteristic Functions; Chapter 3: Bounds for Errors of Normal Approximation; Chapter 4: Asymptotic Expansions Nonlattice Distributions; Chapter 5: Asymptotic Expansions Lattice Distributions; Chapter 6: Two Recent Improvements; Chapter 7: An Application of Stein s Method; Appendix A.1: Random Vectors and Independence; Appendix A.2: Functions of Bounded Variation and Distribution Functions; Appendix A.3: Absolutely Continuous, Singular, and Discrete Probability Measures; Appendix A.4: The Euler-MacLaurin Sum Formula for Functions of Several Variables; References; Index. ... Read more

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Asymptotic methods are frequently used in many branches of both pure and applied mathematics, and this classic text remains the most up-to-date book dealing with one important aspect of this area, namely, asymptotic approximations of integrals. In Asymptotic Approximations of Integrals, all results are proved rigorously, and many of the approximation formulas are accompanied by error bounds. A thorough discussion on multidimensional integrals is given, and references are provided. Asymptotic Approximations of Integrals contains the "distributional method," which is not available elsewhere. Most of the examples in this text come from concrete applications.

Since its publication twelve years ago, significant developments have occurred in the general theory of asymptotic expansions, including smoothing of the Stokes phenomenon, uniform exponentially improved asymptotic expansions, and hyperasymptotics. These new concepts belong to the area now known as "exponential asymptotics." Expositions of these new theories are available in papers published in various journals, but not yet in book form. ... Read more

Customer Reviews (1)

This book is very important to my study in guraduate school.
I am studying "Asyptotic Expantions" in graduate school. To study this field, this book is veryimportant reference one of them. ... Read more

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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

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This book collects up-to-date papers from world experts in a broad variety of relevant applications of approximation theory, including dynamical systems, multiscale modelling of fluid flow, metrology, and geometric modelling to mention a few. The 14 papers in this volume document modern trends in approximation through recent theoretical developments, important computational aspects and multidisciplinary applications. The book is arranged in seven invited surveys, followed by seven contributed research papers. The surveys of the first seven chapters are addressing the following relevant topics: emergent behaviour in large electrical networks, algorithms for multivariate piecewise constant approximation, anisotropic triangulation methods in adaptive image approximation, form assessment in coordinate metrology, discontinuous Galerkin methods for linear problems, a numerical analyst's view of the lattice Boltzmann method, approximation of probability measures on manifolds. Moreover, the diverse contributed papers of the remaining seven chapters reflect recent developments in approximation theory, approximation practice and their applications. Graduate students who wish to discover the state of the art in a number of important directions of approximation algorithms will find this a valuable volume. Established researchers from statisticians through to fluid modellers will find interesting new approaches to solving familiar but challenging problems. This book grew out of the sixth in the conference series on "Algorithms for Approximation", which took place from 31st August to September 4th 2009 in Ambleside in the Lake District of the United Kingdom. ... Read more

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Approximation theory and numerical analysis are central to the creation of accurate computer simulations and mathematical models. Research in these areas can influence the computational techniques used in a variety of mathematical and computational sciences.

This collection of contributed chapters, dedicated to renowned mathematician Gradimir V. Milovanović, represent the recent work of experts in the fields of approximation theory and numerical analysis. These invited contributions describe new trends in these important areas of research including theoretic developments, new computational algorithms, and multidisciplinary applications.

Special features of this volume:

- Presents results and approximation methods in various computational settings including: polynomial and orthogonal systems, analytic functions, and differential equations.

- Provides a historical overview of approximation theory and many of its subdisciplines;

- Contains new results from diverse areas of research spanning mathematics, engineering, and the computational sciences.

"Approximation and Computation" is intended for mathematicians and researchers focusing on approximation theory and numerical analysis, but can also be a valuable resource to students and researchers in the computational and applied sciences.