Introduction To The Theory Of Weighted Polynomial Approximation

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Introduction To The Theory Of Weighted Polynomial Approximation

H N Mhaskar
Introduction To The Theory Of Weighted Polynomial Approximation Book PDF

Author : H N Mhaskar
Publisher : World Scientific
Page : 396 pages
File Size : 52,5 Mb
Release : 1997-01-04
Category : Mathematics
ISBN : 9789814518055

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Introduction To The Theory Of Weighted Polynomial Approximation by H N Mhaskar Pdf

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.

Weighted Polynomial Approximation and Numerical Methods for Integral Equations

Peter Junghanns,Giuseppe Mastroianni,Incoronata Notarangelo
Weighted Polynomial Approximation and Numerical Methods for Integral Equations Book PDF

Author : Peter Junghanns,Giuseppe Mastroianni,Incoronata Notarangelo
Publisher : Springer Nature
Page : 662 pages
File Size : 44,9 Mb
Release : 2021-08-10
Category : Mathematics
ISBN : 9783030774974

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Weighted Polynomial Approximation and Numerical Methods for Integral Equations by Peter Junghanns,Giuseppe Mastroianni,Incoronata Notarangelo Pdf

The book presents a combination of two topics: one coming from the theory of approximation of functions and integrals by interpolation and quadrature, respectively, and the other from the numerical analysis of operator equations, in particular, of integral and related equations. The text focusses on interpolation and quadrature processes for functions defined on bounded and unbounded intervals and having certain singularities at the endpoints of the interval, as well as on numerical methods for Fredholm integral equations of first and second kind with smooth and weakly singular kernel functions, linear and nonlinear Cauchy singular integral equations, and hypersingular integral equations. The book includes both classic and very recent results and will appeal to graduate students and researchers who want to learn about the approximation of functions and the numerical solution of operator equations, in particular integral equations.

Approximation Theory and Methods

M. J. D. Powell
Approximation Theory and Methods Book PDF

Author : M. J. D. Powell
Publisher : Cambridge University Press
Page : 356 pages
File Size : 41,8 Mb
Release : 1981-03-31
Category : Mathematics
ISBN : 0521295149

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Approximation Theory and Methods by M. J. D. Powell Pdf

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.

Fundamentals of Approximation Theory

Hrushikesh Narhar Mhaskar,Devidas V. Pai
Fundamentals of Approximation Theory Book PDF

Author : Hrushikesh Narhar Mhaskar,Devidas V. Pai
Publisher : CRC Press
Page : 580 pages
File Size : 42,5 Mb
Release : 2000
Category : Mathematics
ISBN : 0849309395

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Fundamentals of Approximation Theory by Hrushikesh Narhar Mhaskar,Devidas V. Pai Pdf

The field of approximation theory has become so vast that it intersects with every other branch of analysis and plays an increasingly important role in applications in the applied sciences and engineering. Fundamentals of Approximation Theory presents a systematic, in-depth treatment of some basic topics in approximation theory designed to emphasize the rich connections of the subject with other areas of study. With an approach that moves smoothly from the very concrete to more and more abstract levels, this text provides an outstanding blend of classical and abstract topics. The first five chapters present the core of information that readers need to begin research in this domain. The final three chapters the authors devote to special topics-splined functions, orthogonal polynomials, and best approximation in normed linear spaces- that illustrate how the core material applies in other contexts and expose readers to the use of complex analytic methods in approximation theory. Each chapter contains problems of varying difficulty, including some drawn from contemporary research. Perfect for an introductory graduate-level class, Fundamentals of Approximation Theory also contains enough advanced material to serve more specialized courses at the doctoral level and to interest scientists and engineers.

Exploring Mathematical Analysis, Approximation Theory, and Optimization

Nicholas J. Daras,Michael Th. Rassias,Nikolaos B. Zographopoulos
Exploring Mathematical Analysis Approximation Theory and Optimization Book PDF

Author : Nicholas J. Daras,Michael Th. Rassias,Nikolaos B. Zographopoulos
Publisher : Springer Nature
Page : 474 pages
File Size : 42,9 Mb
Release : 2024-01-04
Category : Mathematics
ISBN : 9783031464874

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Exploring Mathematical Analysis, Approximation Theory, and Optimization by Nicholas J. Daras,Michael Th. Rassias,Nikolaos B. Zographopoulos Pdf

This book compiles research and surveys devoted to the areas of mathematical analysis, approximation theory, and optimization. Being dedicated to A.-M. Legendre's work, contributions to this volume are devoted to those branches of mathematics and its applications that have been influenced, directly or indirectly, by the mathematician. Additional contributions provide a historical background as it relates to Legendre's work and its association to the foundation of Greece's higher education. Topics covered in this book include the investigation of the Jensen-Steffensen inequality, Ostrowski and trapezoid type inequalities, a Hilbert-Type Inequality, Hardy’s inequality, dynamic unilateral contact problems, square-free values of a category of integers, a maximum principle for general nonlinear operators, the application of Ergodic Theory to an alternating series expansion for real numbers, bounds for similarity condition numbers of unbounded operators, finite element methods with higher order polynomials, generating functions for the Fubini type polynomials, local asymptotics for orthonormal polynomials, trends in geometric function theory, quasi variational inclusions, Kleene fixed point theorems, ergodic states, spontaneous symmetry breaking and quasi-averages. It is hoped that this book will be of interest to a wide spectrum of readers from several areas of pure and applied sciences, and will be useful to undergraduate students, graduate level students, and researchers who want to be kept up to date on the results and theories in the subjects covered in this volume.

Limit Theorems of Polynomial Approximation with Exponential Weights

Michael I. Ganzburg,John Rognes
Limit Theorems of Polynomial Approximation with Exponential Weights Book PDF

Author : Michael I. Ganzburg,John Rognes
Publisher : American Mathematical Soc.
Page : 178 pages
File Size : 45,7 Mb
Release : 2008
Category : Approximation theory
ISBN : 9780821840634

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Limit Theorems of Polynomial Approximation with Exponential Weights by Michael I. Ganzburg,John Rognes Pdf

The author develops the limit relations between the errors of polynomial approximation in weighted metrics and apply them to various problems in approximation theory such as asymptotically best constants, convergence of polynomials, approximation of individual functions, and multidimensional limit theorems of polynomial approximation.

Interpolation Processes

Giuseppe Mastroianni,Gradimir Milovanovic
Interpolation Processes Book PDF

Author : Giuseppe Mastroianni,Gradimir Milovanovic
Publisher : Springer Science & Business Media
Page : 452 pages
File Size : 51,9 Mb
Release : 2008-08-24
Category : Mathematics
ISBN : 9783540683490

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Interpolation Processes by Giuseppe Mastroianni,Gradimir Milovanovic Pdf

Interpolation of functions is one of the basic part of Approximation Theory. There are many books on approximation theory, including interpolation methods that - peared in the last fty years, but a few of them are devoted only to interpolation processes. An example is the book of J. Szabados and P. Vértesi: Interpolation of Functions, published in 1990 by World Scienti c. Also, two books deal with a special interpolation problem, the so-called Birkhoff interpolation, written by G.G. Lorentz, K. Jetter, S.D. Riemenschneider (1983) and Y.G. Shi (2003). The classical books on interpolation address numerous negative results, i.e., - sultsondivergentinterpolationprocesses,usuallyconstructedoversomeequidistant system of nodes. The present book deals mainly with new results on convergent - terpolation processes in uniform norm, for algebraic and trigonometric polynomials, not yet published in other textbooks and monographs on approximation theory and numerical mathematics. Basic tools in this eld (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. The rstchapterprovidesanaccountofbasicfactsonapproximationbyalgebraic and trigonometric polynomials introducing the most important concepts on appro- mation of functions. Especially, in Sect. 1.4 we give basic results on interpolation by algebraic polynomials, including representations and computation of interpolation polynomials, Lagrange operators, interpolation errors and uniform convergence in some important classes of functions, as well as an account on the Lebesgue function and some estimates for the Lebesgue constant.

Discrepancy of Signed Measures and Polynomial Approximation

Vladimir V. Andrievskii,Hans-Peter Blatt
Discrepancy of Signed Measures and Polynomial Approximation Book PDF

Author : Vladimir V. Andrievskii,Hans-Peter Blatt
Publisher : Springer Science & Business Media
Page : 444 pages
File Size : 48,9 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9781475749991

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Discrepancy of Signed Measures and Polynomial Approximation by Vladimir V. Andrievskii,Hans-Peter Blatt Pdf

A concise outline of the basic facts of potential theory and quasiconformal mappings makes this book an ideal introduction for non-experts who want to get an idea of applications of potential theory and geometric function theory in various fields of construction analysis.

Theory of Uniform Approximation of Functions by Polynomials

Vladislav K. Dzyadyk,Igor A. Shevchuk
Theory of Uniform Approximation of Functions by Polynomials Book PDF

Author : Vladislav K. Dzyadyk,Igor A. Shevchuk
Publisher : Walter de Gruyter
Page : 497 pages
File Size : 50,5 Mb
Release : 2008-09-25
Category : Mathematics
ISBN : 9783110208245

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Theory of Uniform Approximation of Functions by Polynomials by Vladislav K. Dzyadyk,Igor A. Shevchuk Pdf

A thorough, self-contained and easily accessible treatment of the theory on the polynomial best approximation of functions with respect to maximum norms. The topics include Chebychev theory, Weierstraß theorems, smoothness of functions, and continuation of functions.

Advanced Problems in Constructive Approximation

Martin D. Buhmann,Detlef Mache
Advanced Problems in Constructive Approximation Book PDF

Author : Martin D. Buhmann,Detlef Mache
Publisher : Birkhäuser
Page : 286 pages
File Size : 50,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034876001

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Advanced Problems in Constructive Approximation by Martin D. Buhmann,Detlef Mache Pdf

The current form of modern approximation theory is shaped by many new de velopments which are the subject of this series of conferences. The International Meetings on Approximation Theory attempt to keep track in particular of fun damental advances in the theory of function approximation, for example by (or thogonal) polynomials, (weighted) interpolation, multivariate quasi-interpolation, splines, radial basis functions and several others. This includes both approxima tion order and error estimates, as well as constructions of function systems for approximation of functions on Euclidean spaces and spheres. It is a piece of very good fortune that at all of the IDoMAT meetings, col leagues and friends from all over Europe, and indeed some count ries outside Europe and as far away as China, New Zealand, South Africa and U.S.A. came and dis cussed mathematics at IDoMAT conference facility in Witten-Bommerholz. The conference was, as always, held in a friendly and congenial atmosphere. After each meeting, the delegat es were invited to contribute to the proceed ing's volume, the previous one being published in the same Birkhäuser series as this one. The editors were pleased about the quality of the contributions which could be solicited for the book. They are refereed and we should mention our gratitude to the referees and their work.

Sparse Polynomial Approximation of High-Dimensional Functions

Ben Adcock ,Simone Brugiapaglia,Clayton G. Webster
Sparse Polynomial Approximation of High Dimensional Functions Book PDF

Author : Ben Adcock ,Simone Brugiapaglia,Clayton G. Webster
Publisher : SIAM
Page : 310 pages
File Size : 55,6 Mb
Release : 2022-02-16
Category : Mathematics
ISBN : 9781611976885

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Sparse Polynomial Approximation of High-Dimensional Functions by Ben Adcock ,Simone Brugiapaglia,Clayton G. Webster Pdf

Over seventy years ago, Richard Bellman coined the term “the curse of dimensionality” to describe phenomena and computational challenges that arise in high dimensions. These challenges, in tandem with the ubiquity of high-dimensional functions in real-world applications, have led to a lengthy, focused research effort on high-dimensional approximation—that is, the development of methods for approximating functions of many variables accurately and efficiently from data. This book provides an in-depth treatment of one of the latest installments in this long and ongoing story: sparse polynomial approximation methods. These methods have emerged as useful tools for various high-dimensional approximation tasks arising in a range of applications in computational science and engineering. It begins with a comprehensive overview of best s-term polynomial approximation theory for holomorphic, high-dimensional functions, as well as a detailed survey of applications to parametric differential equations. It then describes methods for computing sparse polynomial approximations, focusing on least squares and compressed sensing techniques. Sparse Polynomial Approximation of High-Dimensional Functions presents the first comprehensive and unified treatment of polynomial approximation techniques that can mitigate the curse of dimensionality in high-dimensional approximation, including least squares and compressed sensing. It develops main concepts in a mathematically rigorous manner, with full proofs given wherever possible, and it contains many numerical examples, each accompanied by downloadable code. The authors provide an extensive bibliography of over 350 relevant references, with an additional annotated bibliography available on the book’s companion website (www.sparse-hd-book.com). This text is aimed at graduate students, postdoctoral fellows, and researchers in mathematics, computer science, and engineering who are interested in high-dimensional polynomial approximation techniques.

An Introduction to the Approximation of Functions

Theodore J. Rivlin
An Introduction to the Approximation of Functions Book PDF

Author : Theodore J. Rivlin
Publisher : Unknown
Page : 168 pages
File Size : 48,7 Mb
Release : 1969
Category : Approximation theory
ISBN : MINN:31951000476079A

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An Introduction to the Approximation of Functions by Theodore J. Rivlin Pdf

Approximation theory is an area of mathematics with important practical applications in computation. This volume provides an introduction to the theoretical foundations which underlie many of the algorithms of everyday use. For each method of approximation studied, at least one algorithm leading to actual numerical approximations is described.

Computational Methods And Function Theory 1997 - Proceedings Of The Third Cmft Conference

Nicolas Papamichael,Stephan Ruscheweyh,E B Saff
Computational Methods And Function Theory 1997 Proceedings Of The Third Cmft Conference Book PDF

Author : Nicolas Papamichael,Stephan Ruscheweyh,E B Saff
Publisher : World Scientific
Page : 666 pages
File Size : 48,5 Mb
Release : 1999-04-14
Category : Electronic
ISBN : 9789814544399

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Computational Methods And Function Theory 1997 - Proceedings Of The Third Cmft Conference by Nicolas Papamichael,Stephan Ruscheweyh,E B Saff Pdf

This volume contains refereed state-of-the-art research articles and extensive surveys on the various aspects of interaction of complex variables and scientific computation as well as on related areas such as function theory and approximation theory.

Advanced Topics In Multivariate Approximation - Proceedings Of The International Workshop

Fontanella F,Jetter Kurt,Laurent P J
Advanced Topics In Multivariate Approximation Proceedings Of The International Workshop Book PDF

Author : Fontanella F,Jetter Kurt,Laurent P J
Publisher : World Scientific
Page : 380 pages
File Size : 52,9 Mb
Release : 1996-11-13
Category : Electronic
ISBN : 9789814547192

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Advanced Topics In Multivariate Approximation - Proceedings Of The International Workshop by Fontanella F,Jetter Kurt,Laurent P J Pdf

This volume consists of 24 refereed carefully edited papers on various topics in multivariate approximation. It represents the proceedings of a workshop organized by the University of Firenze, and held in September 1995 in Montecatini, Italy.The main themes of the volume are multiresolution analysis and wavelets, multidimensional interpolation and smoothing, and computer-aided geometric design. A number of particular topics are included, like subdivision algorithms, constrained approximation and shape-preserving algorithms, thin plate splines, radial basis functions, treatment of scattered data, rational surfaces and offsets, blossoming, grid generation, surface reconstruction, algebraic curves and surfaces, and neural networks.

Multivariate Polynomial Approximation

Manfred Reimer
Multivariate Polynomial Approximation Book PDF

Author : Manfred Reimer
Publisher : Birkhäuser
Page : 361 pages
File Size : 41,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034880954

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Multivariate Polynomial Approximation by Manfred Reimer Pdf

This book introduces general theory by presenting the most important facts on multivariate interpolation, quadrature, orthogonal projections and their summation, all treated under a constructive view, and embedded in the theory of positive linear operators. On this background, the book builds the first comprehensive introduction to the theory of generalized hyperinterpolation. Several parts of the book are based on rotation principles, which are presented in the beginning of the book.