Constructive Approximation

This book presents an in-depth study on advances in constructive approximation theory with recent problems on linear positive operators. State-of-the-art research in constructive approximation is trea

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 volume contains refereed articles originating from the 3rd International Dortmund Meeting on Approximation Theory in Witten-Bommerholz. The contributors are renowned international experts in thei

Lectures on Constructive Approximation: Fourier, Spline, and Wavelet Methods on the Real Line, the Sphere, and the Ball focuses on spherical problems as they occur in the geosciences and medical imagi

Coupled with its sequel, this book gives a connected, unified exposition of Approximation Theory for functions of one real variable. It describes spaces of functions such as Sobolev, Lipschitz, Besov

This monograph presents a broad treatment of developments in an area of constructive approximation involving the so-called "max-product" type operators. The exposition highlights the max-product opera

This two-volume text provides a complete overview of the theory of Banach spaces, emphasising its interplay with classical and harmonic analysis (particularly Sidon sets) and probability. The authors give a full exposition of all results, as well as numerous exercises and comments to complement the text and aid graduate students in functional analysis. The book will also be an invaluable reference volume for researchers in analysis. Volume 1 covers the basics of Banach space theory, operatory theory in Banach spaces, harmonic analysis and probability. The authors also provide an annex devoted to compact Abelian groups. Volume 2 focuses on applications of the tools presented in the first volume, including Dvoretzky's theorem, spaces without the approximation property, Gaussian processes, and more. In volume 2, four leading experts also provide surveys outlining major developments in the field since the publication of the original French edition.

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.

本書《當X世代遇到Z世代的大學教學》探討當前「X世代」的大學教師，如何在面對「Z 世代」時，冒著可能被學生申訴的風險，完成愈來愈艱難的教學任務。 隨著網路科技的快速進展，當前全世界的大學都面臨到這群在學習方式、語言表達、認知思維、價值觀、行為模式、人際關係，甚至情緒管理等，都與以往截然不同的新新人類。面對他們常年活躍於網路世界，大學課堂如何留住他們的「人」與「心」，的確是個新課題。透過作者多年的觀察及教學經驗，經由一些精彩的案例，與讀者分享。 This book attempts to showcase how university teaching is in great transition owing to the unprecedented presence of Gen Z on campus. The author argues that both Gen X and Gen Z need to understand and accept each other, so university teaching can take place in a more effective and constructive way. 上大學到底為了什麼? 現在的大學生有什麼不一樣? 大學教授如何面對網路世代? X世代遇到Z世代會迸出什麼火花?

This textbook offers an accessible introduction to the theory and numerics of approximation methods, combining classical topics of approximation with recent advances in mathematical signal processing,

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 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

Approximation by Multivariate Singular Integrals is the first monograph to illustrate the approximation of multivariate singular integrals to the identity-unit operator. The basic approximation proper

Originally published in 1987, this book is devoted to the approximation of real functions by real rational functions. These are, in many ways, a more convenient tool than polynomials, and interest in them was growing, especially since D. Newman's work in the mid-sixties. The authors aim at presenting the basic achievements of the subject and, for completeness, also discuss some topics from complex rational approximation. Certain classical and modern results from linear approximation theory and spline approximation are also included for comparative purposes. This book will be of value to anyone with an interest in approximation theory and numerical analysis.

This book is intended as a self-contained introduction for non-specialists, or as a reference work for experts, to the particular area of approximation theory that is concerned with exact constants. The results apply mainly to extremal problems in approximation theory, which in turn are closely related to numerical analysis and optimization. The book encompasses a wide range of questions and problems: best approximation by polynomials and splines; linear approximation methods, such as spline-approximation; optimal reconstruction of functions and linear functionals. Many of the results are based on deep facts from analysis and function theory, such as duality theory and comparison theorems; these are presented in chapters 1 and 3. In keeping with the author's intention to make the book as self-contained as possible, chapter 2 contains an introduction to polynomial and spline approximation. Chapters 4 to 7 apply the theory to specific classes of functions. The last chapter deals with n-w

Illustrating the relevance of linear approximation in a variety of fields, Numerical Linear Approximation in C presents a unique collection of linear approximation algorithms that can be used to analy

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.

Dedicated to the well-respected research mathematician Ambikeshwar Sharma, Frontiers in Interpolation and Approximation explores approximation theory, interpolation theory, and classical analysis. Wri

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

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

Approximation Theorems of Mathematical StatisticsThis convenient paperback edition makes a seminal text in statistics accessible to a new generation of students and practitioners. Approximation Theore

This 1999 book is concerned with Diophantine approximation on smooth manifolds embedded in Euclidean space, and its aim is to develop a coherent body of theory comparable with that which already exists for classical Diophantine approximation. In particular, this book deals with Khintchine-type theorems and with the Hausdorff dimension of the associated null sets. After setting out the necessary background material, the authors give a full discussion of Hausdorff dimension and its uses in Diophantine approximation. A wide range of techniques from the number theory arsenal are used to obtain the upper and lower bounds required, and this is an indication of the difficulty of some of the questions considered. The authors go on to consider briefly the p-adic case, and they conclude with a chapter on some applications of metric Diophantine approximation. All researchers with an interest in Diophantine approximation will welcome this book.

The subject of harmonic approximation has recently matured into a coherent research area with extensive applications. This is the first book to give a systematic account of these developments, beginning with classical results concerning uniform approximation on compact sets, and progressing through fusion techniques to deal with approximation on unbounded sets. All the time inspiration is drawn from holomorphic results such as the well-known theorems of Runge and Mergelyan. The final two chapters deal with wide-ranging and surprising applications to the Dirichlet problem, maximum principle, Radon transform and the construction of pathological harmonic functions. This book is aimed at graduate students and researchers who have some knowledge of subharmonic functions, or an interest in holomorphic approximation.

This book is intended as a self-contained introduction for non-specialists, or as a reference work for experts, to the particular area of approximation theory that is concerned with exact constants. The results apply mainly to extremal problems in approximation theory, which in turn are closely related to numerical analysis and optimization. The book encompasses a wide range of questions and problems: best approximation by polynomials and splines; linear approximation methods, such as spline-approximation; optimal reconstruction of functions and linear functionals. Many of the results are based on deep facts from analysis and function theory, such as duality theory and comparison theorems; these are presented in chapters 1 and 3. In keeping with the author's intention to make the book as self-contained as possible, chapter 2 contains an introduction to polynomial and spline approximation. Chapters 4 to 7 apply the theory to specific classes of functions. The last chapter deals with n-w

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.

Originally published in 1987, this book is devoted to the approximation of real functions by real rational functions. These are, in many ways, a more convenient tool than polynomials, and interest in them was growing, especially since D. Newman's work in the mid-sixties. The authors aim at presenting the basic achievements of the subject and, for completeness, also discuss some topics from complex rational approximation. Certain classical and modern results from linear approximation theory and spline approximation are also included for comparative purposes. This book will be of value to anyone with an interest in approximation theory and numerical analysis.

Constructive Living is a Western approach to mental health education based in large part on adaptations of two Japanese psychotherapies, Morita therapy and Naikan therapy. Constructive Living (CL) pre

Using Bishop's work on constructive analysis as a framework, this monograph gives a systematic, detailed and general constructive theory of probability theory and stochastic processes. It is the first extended account of this theory: almost all of the constructive existence and continuity theorems that permeate the book are original. It also contains results and methods hitherto unknown in the constructive and nonconstructive settings. The text features logic only in the common sense and, beyond a certain mathematical maturity, requires no prior training in either constructive mathematics or probability theory. It will thus be accessible and of interest, both to probabilists interested in the foundations of their speciality and to constructive mathematicians who wish to see Bishop's theory applied to a particular field.

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 book presents a systematic overview of approximation by linear combinations of positive linear operators, a useful tool used to increase the order of approximation. Fundamental and recent results

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

Current and historical research methods in approximation theory are presented in this book beginning with the 1800s and following the evolution of approximation theory via the refinement and extension

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

The book discusses the solutions to nonlinear ordinary differential equations (ODEs) using analytical and numerical approximation methods. Recently, analytical approximation methods have been largely

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

Constructive topology is an important new branch of constructive mathematics and logic and this monograph is the first account of constructive topology in book form. Aimed at researchers in mathemati

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

Advances in Applied Mathematics and Approximation Theory: Contributions from AMAT 2012 is a collection of the best articles presented at “Applied Mathematics and Approximation Theory 2012,” an interna

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

Constructive Nonlinear Control presents a broad repertoire of constructive nonlinear designs not available in other works by widening the class of systems and design tools. Several streams of nonlinea