Geometric Approximation Theory

Author : Alexey R. Alimov,Igor’ G. Tsar’kov
Publisher : Springer Nature
Page : 523 pages
File Size : 44,9 Mb
Release : 2022-03-29
Category : Mathematics
ISBN : 9783030909512

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Geometric Approximation Theory by Alexey R. Alimov,Igor’ G. Tsar’kov Pdf

This monograph provides a comprehensive introduction to the classical geometric approximation theory, emphasizing important themes related to the theory including uniqueness, stability, and existence of elements of best approximation. It presents a number of fundamental results for both these and related problems, many of which appear for the first time in monograph form. The text also discusses the interrelations between main objects of geometric approximation theory, formulating a number of auxiliary problems for demonstration. Central ideas include the problems of existence and uniqueness of elements of best approximations as well as properties of sets including subspaces of polynomials and splines, classes of rational functions, and abstract subsets of normed linear spaces. The book begins with a brief introduction to geometric approximation theory, progressing through fundamental classical ideas and results as a basis for various approximation sets, suns, and Chebyshev systems. It concludes with a review of approximation by abstract sets and related problems, presenting novel results throughout the section. This text is suitable for both theoretical and applied viewpoints and especially researchers interested in advanced aspects of the field.

Author : Alexey Alimov,Igor' G. Tsar'kov
Publisher : Unknown
Page : 0 pages
File Size : 46,6 Mb
Release : 2021
Category : Approximation theory
ISBN : 830309095X

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Geometric Approximation Theory by Alexey Alimov,Igor' G. Tsar'kov Pdf

This monograph provides a comprehensive introduction to the classical geometric approximation theory, emphasizing important themes related to the theory including uniqueness, stability, and existence of elements of best approximation. It presents a number of fundamental results for both these and related problems, many of which appear for the first time in monograph form. The text also discusses the interrelations between main objects of geometric approximation theory, formulating a number of auxiliary problems for demonstration. Central ideas include the problems of existence and uniqueness of elements of best approximations as well as properties of sets including subspaces of polynomials and splines, classes of rational functions, and abstract subsets of normed linear spaces. The book begins with a brief introduction to geometric approximation theory, progressing through fundamental classical ideas and results as a basis for various approximation sets, suns, and Chebyshev systems. It concludes with a review of approximation by abstract sets and related problems, presenting novel results throughout the section. This text is suitable for both theoretical and applied viewpoints and especially researchers interested in advanced aspects of the field.

Author : Sariel Har-Peled
Publisher : American Mathematical Soc.
Page : 378 pages
File Size : 53,8 Mb
Release : 2011
Category : Computers
ISBN : 9780821849118

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Geometric Approximation Algorithms by Sariel Har-Peled Pdf

Exact algorithms for dealing with geometric objects are complicated, hard to implement in practice, and slow. Over the last 20 years a theory of geometric approximation algorithms has emerged. These algorithms tend to be simple, fast, and more robust than their exact counterparts. This book is the first to cover geometric approximation algorithms in detail. In addition, more traditional computational geometry techniques that are widely used in developing such algorithms, like sampling, linear programming, etc., are also surveyed. Other topics covered include approximate nearest-neighbor search, shape approximation, coresets, dimension reduction, and embeddings. The topics covered are relatively independent and are supplemented by exercises. Close to 200 color figures are included in the text to illustrate proofs and ideas.

Author : George A. Anastassiou,Sorin G. Gal
Publisher : Springer Science & Business Media
Page : 520 pages
File Size : 51,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461213604

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Approximation Theory by George A. Anastassiou,Sorin G. Gal Pdf

We study in Part I of this monograph the computational aspect of almost all moduli of continuity over wide classes of functions exploiting some of their convexity properties. To our knowledge it is the first time the entire calculus of moduli of smoothness has been included in a book. We then present numerous applications of Approximation Theory, giving exact val ues of errors in explicit forms. The K-functional method is systematically avoided since it produces nonexplicit constants. All other related books so far have allocated very little space to the computational aspect of moduli of smoothness. In Part II, we study/examine the Global Smoothness Preservation Prop erty (GSPP) for almost all known linear approximation operators of ap proximation theory including: trigonometric operators and algebraic in terpolation operators of Lagrange, Hermite-Fejer and Shepard type, also operators of stochastic type, convolution type, wavelet type integral opera tors and singular integral operators, etc. We present also a sufficient general theory for GSPP to hold true. We provide a great variety of applications of GSPP to Approximation Theory and many other fields of mathemat ics such as Functional analysis, and outside of mathematics, fields such as computer-aided geometric design (CAGD). Most of the time GSPP meth ods are optimal. Various moduli of smoothness are intensively involved in Part II. Therefore, methods from Part I can be used to calculate exactly the error of global smoothness preservation. It is the first time in the literature that a book has studied GSPP.

Author : E.B. Safe
Publisher : Elsevier
Page : 506 pages
File Size : 53,6 Mb
Release : 2013-05-09
Category : Mathematics
ISBN : 9780323147774

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Pade and Rational Approximation by E.B. Safe Pdf

Padé and Rational Approximation: Theory and Applications presents the proceedings of the Conference on Rational Approximation with Emphasis on Applications of Padé Approximants, held in Tampa, Florida on December 15-17, 1976. The contributors focus on the interplay of theory, computation, and physical applications. This book is composed of six parts encompassing 44 chapters. The introductory part discusses the general theory of orthogonal polynomials that is the mathematical basis of Padé approximants and related matters evaluation. This text also examines the connection between approximants on a stepline in the ordinary Padé table and certain continued fractions and the convergence of diagonal Padé approximants to a class of functions with an even number of branch points. The following parts deal with the special functions and continued fractions of Padé approximation and the theory of rational approximations. These parts also investigate the geometric convergence of Chebyshev rational approximation on the half line, the optimal approximation by “Almost Classical interpolation, and the incomplete polynomials approximation. The discussion then shifts to the physical applications and computations of the Padé approximants. The concluding part presents the applications of rational approximation to gun fire control and to the White Sands Missile Range Computer Facility. This part also provides a list of some open problems and conjectures concerning polynomials and rational functions. This book is of great benefit to mathematicians, physicists, and laboratory workers.

Author : Marian Neamtu,Larry Schumaker
Publisher : Springer Science & Business Media
Page : 420 pages
File Size : 41,8 Mb
Release : 2011-11-19
Category : Mathematics
ISBN : 9781461407720

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Approximation Theory XIII: San Antonio 2010 by Marian Neamtu,Larry Schumaker Pdf

These proceedings were prepared in connection with the international conference Approximation Theory XIII, which was held March 7–10, 2010 in San Antonio, Texas. The conference was the thirteenth in a series of meetings in Approximation Theory held at various locations in the United States, and was attended by 144 participants. Previous conferences in the series were held in Austin, Texas (1973, 1976, 1980, 1992), College Station, Texas (1983, 1986, 1989, 1995), Nashville, Tennessee (1998), St. Louis, Missouri (2001), Gatlinburg, Tennessee (2004), and San Antonio, Texas (2007). Along with the many plenary speakers, the contributors to this proceedings provided inspiring talks and set a high standard of exposition in their descriptions of new directions for research. Many relevant topics in approximation theory are included in this book, such as abstract approximation, approximation with constraints, interpolation and smoothing, wavelets and frames, shearlets, orthogonal polynomials, univariate and multivariate splines, and complex approximation.

Author : Lloyd N. Trefethen
Publisher : SIAM
Page : 375 pages
File Size : 50,5 Mb
Release : 2019-01-01
Category : Mathematics
ISBN : 9781611975949

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Approximation Theory and Approximation Practice, Extended Edition by Lloyd N. Trefethen Pdf

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 field’s 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 : Narendra Kumar Govil,Ram Mohapatra,Mohammed A. Qazi,Gerhard Schmeisser
Publisher : Springer
Page : 519 pages
File Size : 53,7 Mb
Release : 2017-04-03
Category : Mathematics
ISBN : 9783319492421

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Progress in Approximation Theory and Applicable Complex Analysis by Narendra Kumar Govil,Ram Mohapatra,Mohammed A. Qazi,Gerhard Schmeisser Pdf

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 of classical methods and ending with recent techniques and methodologies. Graduate students, postdocs, and researchers in mathematics, specifically those working in the theory of functions, approximation theory, geometric function theory, and optimization will find new insights as well as a guide to advanced topics. The chapters in this book are grouped into four themes; the first, polynomials (Chapters 1 –8), includes inequalities for polynomials and rational functions, orthogonal polynomials, and location of zeros. The second, inequalities and extremal problems are discussed in Chapters 9 –13. The third, approximation of functions, involves the approximants being polynomials, rational functions, and other types of functions and are covered in Chapters 14 –19. The last theme, quadrature, cubature and applications, comprises the final three chapters and includes an article coauthored by Rahman. This volume serves as a memorial volume to commemorate the distinguished career of Qazi Ibadur Rahman (1934–2013) of the Université de Montréal. Rahman was considered by his peers as one of the prominent experts in analytic theory of polynomials and entire functions. The novelty of his work lies in his profound abilities and skills in applying techniques from other areas of mathematics, such as optimization theory and variational principles, to obtain final answers to countless open problems.

Author : Sorin G. Gal
Publisher : Springer Science & Business Media
Page : 359 pages
File Size : 52,5 Mb
Release : 2010-06-09
Category : Mathematics
ISBN : 9780817647032

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Shape-Preserving Approximation by Real and Complex Polynomials by Sorin G. Gal Pdf

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

Author : Themistocles M. Rassias,Vijay Gupta
Publisher : Springer
Page : 741 pages
File Size : 55,5 Mb
Release : 2016-06-03
Category : Mathematics
ISBN : 9783319312811

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Mathematical Analysis, Approximation Theory and Their Applications by Themistocles M. Rassias,Vijay Gupta Pdf

Designed for graduate students, researchers, and engineers in mathematics, optimization, and economics, this self-contained volume presents theory, methods, and applications in mathematical analysis and approximation theory. Specific topics include: approximation of functions by linear positive operators with applications to computer aided geometric design, numerical analysis, optimization theory, and solutions of differential equations. Recent and significant developments in approximation theory, special functions and q-calculus along with their applications to mathematics, engineering, and social sciences are discussed and analyzed. Each chapter enriches the understanding of current research problems and theories in pure and applied research.

Author : Vijay Gupta,Ravi P. Agarwal
Publisher : Springer Science & Business Media
Page : 368 pages
File Size : 53,7 Mb
Release : 2014-01-08
Category : Mathematics
ISBN : 9783319027654

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Convergence Estimates in Approximation Theory by Vijay Gupta,Ravi P. Agarwal Pdf

The study of linear positive operators is an area of mathematical studies with significant relevance to studies of computer-aided geometric design, numerical analysis, and differential equations. This book focuses on the convergence of linear positive operators in real and complex domains. The theoretical aspects of these operators have been an active area of research over the past few decades. In this volume, authors Gupta and Agarwal explore new and more efficient methods of applying this research to studies in Optimization and Analysis. The text will be of interest to upper-level students seeking an introduction to the field and to researchers developing innovative approaches.

Author : Kurt Jetter,Martin Buhmann,Werner Haussmann,Robert Schaback,Joachim Stoeckler
Publisher : Elsevier
Page : 357 pages
File Size : 43,9 Mb
Release : 2005-11-15
Category : Mathematics
ISBN : 9780080462042

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Topics in Multivariate Approximation and Interpolation by Kurt Jetter,Martin Buhmann,Werner Haussmann,Robert Schaback,Joachim Stoeckler Pdf

This book is a collection of eleven articles, written by leading experts and dealing with special topics in Multivariate Approximation and Interpolation. The material discussed here has far-reaching applications in many areas of Applied Mathematics, such as in Computer Aided Geometric Design, in Mathematical Modelling, in Signal and Image Processing and in Machine Learning, to mention a few. The book aims at giving a comprehensive information leading the reader from the fundamental notions and results of each field to the forefront of research. It is an ideal and up-to-date introduction for graduate students specializing in these topics, and for researchers in universities and in industry. A collection of articles of highest scientific standard An excellent introduction and overview of recent topics from multivariate approximation A valuable source of references for specialists in the field A representation of the state-of-the-art in selected areas of multivariate approximation A rigorous mathematical introduction to special topics of interdisciplinary research

Author : C. K. Chui,L. L. Schumaker,F.I. Utreras
Publisher : Elsevier
Page : 346 pages
File Size : 45,9 Mb
Release : 2014-05-10
Category : Mathematics
ISBN : 9781483271002

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Topics in Multivariate Approximation by C. K. Chui,L. L. Schumaker,F.I. Utreras Pdf

Topics in Multivariate Approximation contains the proceedings of an international workshop on multivariate approximation held at the University of Chile in Santiago, Chile, on December 15-19, 1986. Leading researchers in the field discussed several problem areas related to multivariate approximation and tackled topics ranging from multivariate splines and fitting of scattered data to tensor approximation methods and multivariate polynomial approximation. Numerical grid generation and finite element methods were also explored, along with constrained interpolation and smoothing. Comprised of 22 chapters, this book first describes the application of Boolean methods of approximation in combination with the theory of right invertible operators to bivariate Fourier expansions. The reader is then introduced to ill-posed problems in multivariate approximation; interpolation of scattered data by radial functions; and shape-preserving surface interpolation. Subsequent chapters focus on approximation by harmonic functions; numerical generation of nested series of general triangular grids; triangulation methods; and inequalities arising from best local approximations in rectangles. A bibliography of multivariate approximation concludes the book. This monograph will be of interest to mathematicians.

Author : Gregory E. Fasshauer,Larry L. Schumaker
Publisher : Unknown
Page : 395 pages
File Size : 42,5 Mb
Release : 2014
Category : Approximation theory
ISBN : 3319064053

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Approximation Theory XIV by Gregory E. Fasshauer,Larry L. Schumaker Pdf

Author : Charles K. Chui
Publisher : World Scientific
Page : 606 pages
File Size : 45,8 Mb
Release : 1995
Category : Mathematics
ISBN : 9789814532594

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Approximation Theory VIII by Charles K. Chui Pdf

This is the collection of the refereed and edited papers presented at the 8th Texas International Conference on Approximation Theory. It is interdisciplinary in nature and consists of two volumes. The central theme of Vol. I is the core of approximation theory. It includes such important areas as qualitative approximations, interpolation theory, rational approximations, radial-basis functions, and splines. The second volume focuses on topics related to wavelet analysis, including multiresolution and multi-level approximation, subdivision schemes in CAGD, and applications.