Bernstein Polynomials
Bernstein Polynomials

Author: G. G. Lorentz

Publisher: American Mathematical Soc.

Published: 2012

Total Pages: 146

ISBN-13: 0821875582

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Bernstein polynomials are a remarkable family of polynomials associated to any given function on the unit interval. Their first notable appearance was in Bernstein's proof of the Weierstrass approximation theorem. This book gives an exhaustive exposition of the main facts about the Bernstein polynomials and discusses some of their applications in analysis. The first three chapters of the book give an introduction to a theory of singular integrals by means of the particular instance of Bernstein polynomials. The author writes in the preface to this second edition, "After the trigonometric integrals, Bernstein polynomials are the most important and interesting concrete operators on a space of continuous functions. Since the appearance of the first edition of this book [in 1953], the interest in this subject has continued. In an appendix we have summed up a few of the most important papers that have appeared since."

Interpolation and Approximation by Polynomials
Interpolation and Approximation by Polynomials

Author: George M. Phillips

Publisher: Springer Science & Business Media

Published: 2006-04-06

Total Pages: 325

ISBN-13: 0387216820

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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.

Spline Functions on Triangulations
Spline Functions on Triangulations

Author: Ming-Jun Lai

Publisher: Cambridge University Press

Published: 2007-04-19

Total Pages: 28

ISBN-13: 0521875927

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Comprehensive graduate text offering a detailed mathematical treatment of polynomial splines on triangulations.

Bernstein Operators and Their Properties
Bernstein Operators and Their Properties

Author: Jorge Bustamante

Publisher: Birkhäuser

Published: 2017-04-13

Total Pages: 423

ISBN-13: 3319554026

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This book provides comprehensive information on the main aspects of Bernstein operators, based on the literature to date. Bernstein operators have a long-standing history and many papers have been written on them. Among all types of positive linear operators, they occupy a unique position because of their elegance and notable approximation properties. This book presents carefully selected material from the vast body of literature on this topic. In addition, it highlights new material, including several results (with proofs) appearing in a book for the first time. To facilitate comprehension, exercises are included at the end of each chapter. The book is largely self-contained and the methods in the proofs are kept as straightforward as possible. Further, it requires only a basic grasp of analysis, making it a valuable and appealing resource for advanced graduate students and researchers alike.

Extremal Problems and Inequalities of Markov-Bernstein Type for Algebraic Polynomials
Extremal Problems and Inequalities of Markov-Bernstein Type for Algebraic Polynomials

Author: Robert B. Gardner

Publisher: Elsevier

Published: 2022-02-15

Total Pages: 442

ISBN-13: 0128119888

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Bernstein-type Inequalities for Polynomials and Rational Functions is an integrated, powerful and clear presentation of the emergent field in approximation theory. It presents a unified description of solution norms relevant to complex polynomials, rational functions and exponential functions. Primarily for graduate students and first year PhDs, this book is useful for any researcher exploring problems which require derivative estimates. It is particularly useful for those studying inverse problems in approximation theory. Applies Bernstein-type Inequalities to any problem where derivative estimates are necessary Presents complex math in a clean and simple way, progressing readers from polynomials into rational functions Contains exhaustive references with thousands of citations to articles and books Features methods to solve inverse problems across approximation theory Includes open problems for further research

Numerical Methods in Economics
Numerical Methods in Economics

Author: Kenneth L. Judd

Publisher: MIT Press

Published: 1998-09-28

Total Pages: 662

ISBN-13: 9780262100717

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To harness the full power of computer technology, economists need to use a broad range of mathematical techniques. In this book, Kenneth Judd presents techniques from the numerical analysis and applied mathematics literatures and shows how to use them in economic analyses. The book is divided into five parts. Part I provides a general introduction. Part II presents basics from numerical analysis on R^n, including linear equations, iterative methods, optimization, nonlinear equations, approximation methods, numerical integration and differentiation, and Monte Carlo methods. Part III covers methods for dynamic problems, including finite difference methods, projection methods, and numerical dynamic programming. Part IV covers perturbation and asymptotic solution methods. Finally, Part V covers applications to dynamic equilibrium analysis, including solution methods for perfect foresight models and rational expectation models. A website contains supplementary material including programs and answers to exercises.

Numerical Analysis
Numerical Analysis

Author: Andrew R. Mitchell

Publisher: World Scientific

Published: 1996

Total Pages: 384

ISBN-13: 9789810227197

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This volume is intended to mark the 75th birthday of A R Mitchell, of the University of Dundee. It consists of a collection of articles written by numerical analysts having links with Ron Mitchell, as colleagues, collaborators, former students, or as visitors to Dundee. Ron Mitchell is known for his books and articles contributing to the numerical analysis of partial differential equations; he has also made major contributions to the development of numerical analysis in the UK and abroad, and his many human qualitites are such that he is held in high regard and looked on with great affection by the numerical analysis community. The list of contributors is evidence of the esteem in which he is held, and of the way in which his influence has spread through his former students and fellow workers. In addition to contributions relevant to his own specialist subjects, there are also papers on a wide range of subjects in numerical analysis.

Shape-Preserving Approximation by Real and Complex Polynomials
Shape-Preserving Approximation by Real and Complex Polynomials

Author: Sorin G. Gal

Publisher: Springer Science & Business Media

Published: 2010-06-09

Total Pages: 359

ISBN-13: 0817647031

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First comprehensive treatment in book form of shape-preserving approximation by real or complex polynomials in one or several variables Of interest to grad students and researchers in approximation theory, mathematical analysis, numerical analysis, Computer Aided Geometric Design, robotics, data fitting, chemistry, fluid mechanics, and engineering Contains many open problems to spur future research Rich and updated bibliography

Over and Over Again
Over and Over Again

Author: Gengzhe Chang

Publisher: Cambridge University Press

Published: 1997

Total Pages: 328

ISBN-13: 9780883856413

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Mathematical theme that relates chaos, graphics and geometry, all with just high school maths background.

Constructive Approximation
Constructive Approximation

Author: Ronald A. DeVore

Publisher: Springer Science & Business Media

Published: 1993-11-04

Total Pages: 468

ISBN-13: 9783540506270

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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 rearrangement-invariant function spaces and interpolation of operators. Other topics include Weierstrauss and best approximation theorems, properties of polynomials and splines. It contains history and proofs with an emphasis on principal results.