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Geometric Approximation Theory

  • Book
  • © 2021

Overview

Authors:
  1. Alexey R. Alimov

    1. Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russia

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    You can also search for this author in PubMed   Google Scholar

  2. Igor’ G. Tsar’kov

    1. Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russia

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    You can also search for this author in PubMed   Google Scholar

  • Presents novel results in monograph form
  • Funded by the Russian Foundation for Basic Research
  • Suitable for researchers and postgraduates

Part of the book series: Springer Monographs in Mathematics (SMM)

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Table of contents (17 chapters)

  1. Front Matter

    Pages i-xxi
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  2. Main Notation, Definitions, Auxiliary Results, and Examples

    • Alexey R. Alimov, Igor’ G. Tsar’kov
    Pages 1-18

  3. Chebyshev Alternation Theorem. Haar’s and Mairhuber’s Theorems

    • Alexey R. Alimov, Igor’ G. Tsar’kov
    Pages 19-46

  4. Best Approximation in Euclidean Spaces

    • Alexey R. Alimov, Igor’ G. Tsar’kov
    Pages 47-57

  5. Existence. Compact, Boundedly Compact, Approximatively Compact, and \(\tau \)-Compact Sets. Continuity of the Metric Projection

    • Alexey R. Alimov, Igor’ G. Tsar’kov
    Pages 59-78

  6. Characterization of Best Approximation and Solar Properties of Sets

    • Alexey R. Alimov, Igor’ G. Tsar’kov
    Pages 79-94

  7. Convexity of Chebyshev Sets and Suns

    • Alexey R. Alimov, Igor’ G. Tsar’kov
    Pages 95-114

  8. Connectedness and Approximative Properties of Sets. Stability of the Metric Projection and Its Relation to Other Approximative Properties

    • Alexey R. Alimov, Igor’ G. Tsar’kov
    Pages 115-156

  9. Existence of Chebyshev Subspaces

    • Alexey R. Alimov, Igor’ G. Tsar’kov
    Pages 157-166

  10. Efimov–Stechkin Spaces. Uniform Convexity and Uniform Smoothness. Uniqueness and Strong Uniqueness of Best Approximation in Uniformly Convex Spaces

    • Alexey R. Alimov, Igor’ G. Tsar’kov
    Pages 167-186

  11. Solarity of Chebyshev Sets

    • Alexey R. Alimov, Igor’ G. Tsar’kov
    Pages 187-205

  12. Rational Approximation

    • Alexey R. Alimov, Igor’ G. Tsar’kov
    Pages 207-228

  13. Haar Cones and Varisolvency

    • Alexey R. Alimov, Igor’ G. Tsar’kov
    Pages 229-242

  14. Approximation of Vector-Valued Functions

    • Alexey R. Alimov, Igor’ G. Tsar’kov
    Pages 243-262

  15. The Jung Constant

    • Alexey R. Alimov, Igor’ G. Tsar’kov
    Pages 263-294

  16. Chebyshev Centre of a Set. The Problem of Simultaneous Approximation of a Class by a Singleton Set

    • Alexey R. Alimov, Igor’ G. Tsar’kov
    Pages 295-363

  17. Width. Approximation by a Family of Sets

    • Alexey R. Alimov, Igor’ G. Tsar’kov
    Pages 365-416

  18. Approximative Properties of Arbitrary Sets in Normed Linear Spaces. Almost Chebyshev Sets and Sets of Almost Uniqueness

    • Alexey R. Alimov, Igor’ G. Tsar’kov
    Pages 417-458

  19. Back Matter

    Pages 459-508
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Keywords

About this book

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. Itconcludes 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. 

Authors and Affiliations

  • Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russia

    Alexey R. Alimov, Igor’ G. Tsar’kov

Bibliographic Information

  • Book Title: Geometric Approximation Theory

  • Authors: Alexey R. Alimov, Igor’ G. Tsar’kov

  • Series Title: Springer Monographs in Mathematics

  • DOI: https://doi.org/10.1007/978-3-030-90951-2

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021

  • Hardcover ISBN: 978-3-030-90950-5Published: 29 March 2022

  • Softcover ISBN: 978-3-030-90953-6Published: 30 March 2023

  • eBook ISBN: 978-3-030-90951-2Published: 29 March 2022

  • Series ISSN: 1439-7382

  • Series E-ISSN: 2196-9922

  • Edition Number: 1

  • Number of Pages: XXI, 508

  • Number of Illustrations: 21 b/w illustrations

  • Topics: Approximations and Expansions

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