Approximation

This book provides a systematic survey of classical and recent results on hyperbolic cross approximation. Motivated by numerous applications, the last two decades have seen great success in studying m

This self-contained, systematic treatment of multivariate approximation begins with classical linear approximation, and moves on to contemporary nonlinear approximation. It covers substantial new developments in the linear approximation theory of classes with mixed smoothness, and shows how it is directly related to deep problems in other areas of mathematics. For example, numerical integration of these classes is closely related to discrepancy theory and to nonlinear approximation with respect to special redundant dictionaries, and estimates of the entropy numbers of classes with mixed smoothness are closely related to (in some cases equivalent to) the Small Ball Problem from probability theory. The useful background material included in the book makes it accessible to graduate students. Researchers will find that the many open problems in the theory outlined in the book provide helpful directions and guidance for their own research in this exciting and active area.

This second edition includes eleven new sections based on the approximation of matrix functions, deflating the solution space and improving the accuracy of approximate solutions, iterative solution of

In this monograph, we present the authors' recent work of the last seven years in Approximation Theory. Chapters are self-contained and can be read independently and advanced courses can be taught out

This book is the first one that gives a systematic presentation of recent fundamental results on greedy approximation with respect to bases. Greedy approximation is a very fresh area of research where

This book is aimed at the approximation of set-valued functions with compact sets in an Euclidean space as values. The interest in set-valued functions is rather new. Such functions arise in various m

This monograph deals with the quantitative overconvergence phenomenon in complex approximation by various operators. The book is divided into three chapters. First, the results for the Schurer-Faber o

The approximation of a continuous function by either an algebraic polynomial, a trigonometric polynomial, or a spline, is an important issue in application areas like computer-aided geometric design a

This and the earlier book by R.A. DeVore and G.G. Lorentz (Vol. 303 of the same series), cover the whole field of approximation of functions of one real variable. The main subject of this volume is ap

This book contains an exposition of several results related with direct and converse theorems in the theory of approximation by algebraic polynomials in a finite interval. In addition, some facts conc

This first book on greedy approximation gives a systematic presentation of the fundamental results. It also contains an introduction to two hot topics in numerical mathematics: learning theory and compressed sensing. Nonlinear approximation is becoming increasingly important, especially since two types are frequently employed in applications: adaptive methods are used in PDE solvers, while m-term approximation is used in image/signal/data processing, as well as in the design of neural networks. The fundamental question of nonlinear approximation is how to devise good constructive methods (algorithms) and recent results have established that greedy type algorithms may be the solution. The author has drawn on his own teaching experience to write a book ideally suited to graduate courses. The reader does not require a broad background to understand the material. Important open problems are included to give students and professionals alike ideas for further research.

This book presents numerical and other approximation techniques for solving various types of mathematical problems that cannot be solved analytically. In addition to well known methods, it contains

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 use

Many practical applications require the reconstruction of a multivariate function from discrete, unstructured data. This book gives a self-contained, complete introduction into this subject. It concentrates on truly meshless methods such as radial basis functions, moving least squares, and partitions of unity. The book starts with an overview on typical applications of scattered data approximation, coming from surface reconstruction, fluid-structure interaction, and the numerical solution of partial differential equations. It then leads the reader from basic properties to the current state of research, addressing all important issues, such as existence, uniqueness, approximation properties, numerical stability, and efficient implementation. Each chapter ends with a section giving information on the historical background and hints for further reading. Complete proofs are included, making this perfectly suited for graduate courses on multivariate approximation and it can be used to suppo

In this Spanish bestseller, now available in English for the first time, the famous Biblical scholar Jose Antonio Pagola reconstructs the historical Jesus with a scholarly exegetical and theological

This monograph is concerned with the qualitative theory of best L1-approximation from finite-dimensional subspaces. It presents a survey of recent research that extends 'classical' results concerned with best uniform approximation to the L1 case. The work is organized in such a way as to be useful for self-study or as a text for advanced courses. It begins with a basic introduction to the concepts of approximation theory before addressing one- or two-sided best approximation from finite-dimensional subspaces and approaches to the computation of these. At the end of each chapter is a series of exercises; these give the reader an opportunity to test understanding and also contain some theoretical digressions and extensions of the text.

Algebraic numbers can approximate and classify any real number. Here, the author gathers together results about such approximations and classifications. Written for a broad audience, the book is accessible and self-contained, with complete and detailed proofs. Starting from continued fractions and Khintchine's theorem, Bugeaud introduces a variety of techniques, ranging from explicit constructions to metric number theory, including the theory of Hausdorff dimension. So armed, the reader is led to such celebrated advanced results as the proof of Mahler's conjecture on S-numbers, the Jarnik–Besicovitch theorem, and the existence of T-numbers. Brief consideration is given both to the p-adic and the formal power series cases. Thus the book can be used for graduate courses on Diophantine approximation (some 40 exercises are supplied), or as an introduction for non-experts. Specialists will appreciate the collection of over 50 open problems and the rich and comprehensive list of more than 60

A powerful tool for the approximate solution of differential equations, the finite element is extensively used in industry and research. This book offers students of engineering and physics a comprehe

Many practical applications require the reconstruction of a multivariate function from discrete, unstructured data. This book gives a self-contained, complete introduction into this subject. It concentrates on truly meshless methods such as radial basis functions, moving least squares, and partitions of unity. The book starts with an overview on typical applications of scattered data approximation, coming from surface reconstruction, fluid-structure interaction, and the numerical solution of partial differential equations. It then leads the reader from basic properties to the current state of research, addressing all important issues, such as existence, uniqueness, approximation properties, numerical stability, and efficient implementation. Each chapter ends with a section giving information on the historical background and hints for further reading. Complete proofs are included, making this perfectly suited for graduate courses on multivariate approximation and it can be used to suppo

Algebraic numbers can approximate and classify any real number. Here, the author gathers together results about such approximations and classifications. Written for a broad audience, the book is accessible and self-contained, with complete and detailed proofs. Starting from continued fractions and Khintchine's theorem, Bugeaud introduces a variety of techniques, ranging from explicit constructions to metric number theory, including the theory of Hausdorff dimension. So armed, the reader is led to such celebrated advanced results as the proof of Mahler's conjecture on S-numbers, the Jarnik–Besicovitch theorem, and the existence of T-numbers. Brief consideration is given both to the p-adic and the formal power series cases. Thus the book can be used for graduate courses on Diophantine approximation (some 40 exercises are supplied), or as an introduction for non-experts. Specialists will appreciate the collection of over 50 open problems and the rich and comprehensive list of more than 60

Stochastic approximation is a relatively new technique for studying the properties of an experimental situation; it has important applications in fields such as medicine and engineering. The subject can be treated either largely as a branch of pure mathematics, or else from an empirical and practical angle. In this book, Dr Wasan gives a rigorous mathematical treatment of the subject, drawing together the scattered results of a number of authors. He discusses the conditions under which the method gives a valid approximation to the required solution; methods for optimal choice of parameters to hasten convergence; the comparison of the method with other techniques. The discussion and proofs of theorems are given in enough detail to make them easy to follow, while a number of interesting examples show how the techniques may be applied in many fields.

This concisely written book gives an elementary introduction to a classical area of mathematics - approximation theory - in a way that naturally leads to the modern field of wavelets. The ex

A pioneer of many modern developments in approximation theory, Achieser begins this text with approximation problems in linear normalized spaces and the ideas of P. L. Tchebysheff. He then examines th

This classic text remains the most up-to-date book to deal with asymptotic approximations of integrals. All results discussed are proved rigorously, and many of the approximation formulas are accompan

Covering the basic techniques used in the latest research work, the author consolidates progress made so far, including some very recent and promising results, and conveys the beauty and excitement of

Covering the basic techniques used in the latest research work, the author consolidates progress made so far, including some very recent and promising results, and conveys the beauty and excitement of