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Author : Theodore J. Rivlin
Publisher : Courier Corporation
Page : 164 pages
File Size : 46,5 Mb
Release : 1981-01-01
Category : Mathematics
ISBN : 0486640698
Mathematics of Computing -- Numerical Analysis.
Author : Lloyd N. Trefethen
Publisher : SIAM
Page : 375 pages
File Size : 55,6 Mb
Release : 2019-01-01
Category : Mathematics
ISBN : 9781611975949
This is a textbook on classical polynomial and rational approximation theory for the twenty-first century. Aimed at advanced undergraduates and graduate students across all of applied mathematics, it uses MATLAB to teach the fields most important ideas and results. Approximation Theory and Approximation Practice, Extended Edition differs fundamentally from other works on approximation theory in a number of ways: its emphasis is on topics close to numerical algorithms; concepts are illustrated with Chebfun; and each chapter is a PUBLISHable MATLAB M-file, available online. The book centers on theorems and methods for analytic functions, which appear so often in applications, rather than on functions at the edge of discontinuity with their seductive theoretical challenges. Original sources are cited rather than textbooks, and each item in the bibliography is accompanied by an editorial comment. In addition, each chapter has a collection of exercises, which span a wide range from mathematical theory to Chebfun-based numerical experimentation. This textbook is appropriate for advanced undergraduate or graduate students who have an understanding of numerical analysis and complex analysis. It is also appropriate for seasoned mathematicians who use MATLAB.
Author : M. J. D. Powell
Publisher : Cambridge University Press
Page : 356 pages
File Size : 43,5 Mb
Release : 1981-03-31
Category : Mathematics
ISBN : 0521295149
Most functions that occur in mathematics cannot be used directly in computer calculations. Instead they are approximated by manageable functions such as polynomials and piecewise polynomials. The general theory of the subject and its application to polynomial approximation are classical, but piecewise polynomials have become far more useful during the last twenty years. Thus many important theoretical properties have been found recently and many new techniques for the automatic calculation of approximations to prescribed accuracy have been developed. This book gives a thorough and coherent introduction to the theory that is the basis of current approximation methods. Professor Powell describes and analyses the main techniques of calculation supplying sufficient motivation throughout the book to make it accessible to scientists and engineers who require approximation methods for practical needs. Because the book is based on a course of lectures to third-year undergraduates in mathematics at Cambridge University, sufficient attention is given to theory to make it highly suitable as a mathematical textbook at undergraduate or postgraduate level.
Author : H N Mhaskar
Publisher : World Scientific
Page : 396 pages
File Size : 50,7 Mb
Release : 1997-01-04
Category : Mathematics
ISBN : 9789814518055
In this book, we have attempted to explain a variety of different techniques and ideas which have contributed to this subject in its course of successive refinements during the last 25 years. There are other books and surveys reviewing the ideas from the perspective of either potential theory or orthogonal polynomials. The main thrust of this book is to introduce the subject from an approximation theory point of view. Thus, the main motivation is to study analogues of results from classical trigonometric approximation theory, introducing other ideas as needed. It is not our objective to survey the most recent results, but merely to introduce to the readers the thought processes and ideas as they are developed.This book is intended to be self-contained, although the reader is expected to be familiar with rudimentary real and complex analysis. It will also help to have studied elementary trigonometric approximation theory, and have some exposure to orthogonal polynomials.
Author : Elliott Ward Cheney,William Allan Light
Publisher : American Mathematical Soc.
Page : 379 pages
File Size : 55,5 Mb
Release : 2009-01-13
Category : Mathematics
ISBN : 9780821847985
This textbook is designed for graduate students in mathematics, physics, engineering, and computer science. Its purpose is to guide the reader in exploring contemporary approximation theory. The emphasis is on multi-variable approximation theory, i.e., the approximation of functions in several variables, as opposed to the classical theory of functions in one variable. Most of the topics in the book, heretofore accessible only through research papers, are treated here from the basics to the currently active research, often motivated by practical problems arising in diverse applications such as science, engineering, geophysics, and business and economics. Among these topics are projections, interpolation paradigms, positive definite functions, interpolation theorems of Schoenberg and Micchelli, tomography, artificial neural networks, wavelets, thin-plate splines, box splines, ridge functions, and convolutions. An important and valuable feature of the book is the bibliography of almost 600 items directing the reader to important books and research papers. There are 438 problems and exercises scattered through the book allowing the student reader to get a better understanding of the subject.
Author : A. F. Timan
Publisher : Elsevier
Page : 644 pages
File Size : 43,9 Mb
Release : 2014-07-22
Category : Mathematics
ISBN : 9781483184814
Theory of Approximation of Functions of a Real Variable discusses a number of fundamental parts of the modern theory of approximation of functions of a real variable. The material is grouped around the problem of the connection between the best approximation of functions to their structural properties. This text is composed of eight chapters that highlight the relationship between the various structural properties of real functions and the character of possible approximations to them by polynomials and other functions of simple construction. Each chapter concludes with a section containing various problems and theorems, which supplement the main text. The first chapters tackle the Weierstrass's theorem, the best approximation by polynomials on a finite segment, and some compact classes of functions and their structural properties. The subsequent chapters describe some properties of algebraic polynomials and transcendental integral functions of exponential type, as well as the direct theorems of the constructive theory of functions. These topics are followed by discussions of differential and constructive characteristics of converse theorems. The final chapters explore other theorems connecting the best approximations functions with their structural properties. These chapters also deal with the linear processes of approximation of functions by polynomials. The book is intended for post-graduate students and for mathematical students taking advanced courses, as well as to workers in the field of the theory of functions.
Author : Ben Adcock,Simone Brugiapaglia,Clayton G. Webster
Publisher : Society for Industrial and Applied Mathematics (SIAM)
Page : 0 pages
File Size : 42,5 Mb
Release : 2021
Category : Approximation theory
ISBN : 1611976871
"This is a book about polynomial approximation in high dimensions"--
Author : G. G. Lorentz
Publisher : Unknown
Page : 188 pages
File Size : 51,9 Mb
Release : 1986
Category : Electronic
ISBN : OCLC:797556999
Author : Kendall Atkinson,Weimin Han
Publisher : Springer Science & Business Media
Page : 253 pages
File Size : 54,5 Mb
Release : 2012-02-17
Category : Mathematics
ISBN : 9783642259821
These notes provide an introduction to the theory of spherical harmonics in an arbitrary dimension as well as an overview of classical and recent results on some aspects of the approximation of functions by spherical polynomials and numerical integration over the unit sphere. The notes are intended for graduate students in the mathematical sciences and researchers who are interested in solving problems involving partial differential and integral equations on the unit sphere, especially on the unit sphere in three-dimensional Euclidean space. Some related work for approximation on the unit disk in the plane is also briefly discussed, with results being generalizable to the unit ball in more dimensions.
Author : Vladislav Kirillovich Dzi︠a︡dyk,Igor A. Shevchuk
Publisher : de Gruyter
Page : 504 pages
File Size : 54,5 Mb
Release : 2008
Category : MATHEMATICS
ISBN : UCSC:32106020000490
Review text: "The book could be of interest for all who work in approximation theory and related fields; it should not be overlooked by university libraries."In: Ems Newsletter 3/2009 "It is useful for students interested in uniform approximation theory, and it can be used as a reference book for researchers as well."In: L'Enseignement Mathématique 2/2008.
Author : Rivlin
Publisher : Unknown
Page : 128 pages
File Size : 54,7 Mb
Release : 1969-06-01
Category : Electronic
ISBN : 0471004669
Author : Nikolaĭ Pavlovich Korneĭchuk
Publisher : Cambridge University Press
Page : 472 pages
File Size : 55,9 Mb
Release : 1991-06-06
Category : Mathematics
ISBN : 0521382343
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-widths and generalises some of the ideas of the earlier chapters. Each chapter concludes with commentary, exercises and extensions of results. A substantial bibliography is included. Many of the results collected here have not been gathered together in book form before, so it will be essential reading for approximation theorists.
Author : Carl De Boor,American Mathematical Society
Publisher : American Mathematical Soc.
Page : 152 pages
File Size : 46,8 Mb
Release : 1986-12-31
Category : Mathematics
ISBN : 0821867431
The papers in this book, first presented at a 1986 AMS Short Course, give a brief introduction to approximation theory and some of its current areas of active research, both theoretical and applied. The first lecture describes and illustrates the basic concerns of the field. Topics highlighted in the other lectures include the following: approximation in the complex domain, $N$-width, optimal recovery, interpolation, algorithms for approximation, and splines, with a strong emphasis on a multivariate setting for the last three topics. The book is aimed at mathematicians interested in an introduction to areas of current research and to engineers and scientists interested in exploring the field for possible applications to their own fields. The book is best understood by those with a standard first graduate course in real and complex analysis, but some of the presentations are accessible with the minimal requirements of advanced calculus and linear algebra.
Author : Elliott Ward Cheney
Publisher : Chelsea Publishing Company, Incorporated
Page : 280 pages
File Size : 40,7 Mb
Release : 1982
Category : Mathematics
ISBN : STANFORD:36105033183943
Author : Johan De Villiers
Publisher : Springer Science & Business Media
Page : 406 pages
File Size : 44,6 Mb
Release : 2012-06-30
Category : Mathematics
ISBN : 9789491216503
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 and signal analysis. This book is an introduction to the mathematical analysis of such approximation, and, with the prerequisites of only calculus and linear algebra, the material is targeted at senior undergraduate level, with a treatment that is both rigorous and self-contained. The topics include polynomial interpolation; Bernstein polynomials and the Weierstrass theorem; best approximations in the general setting of normed linear spaces and inner product spaces; best uniform polynomial approximation; orthogonal polynomials; Newton-Cotes , Gauss and Clenshaw-Curtis quadrature; the Euler-Maclaurin formula ; approximation of periodic functions; the uniform convergence of Fourier series; spline approximation,with an extensive treatment of local spline interpolation,and its application in quadrature. Exercises are provided at the end of each chapter
