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Selected Topics in Approximation and Computation$
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Marek A. Kowalski, Krzystof A. Sikorski, and Frank Stenger

Print publication date: 1995

Print ISBN-13: 9780195080599

Published to Oxford Scholarship Online: November 2020

DOI: 10.1093/oso/9780195080599.001.0001

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PRINTED FROM OXFORD SCHOLARSHIP ONLINE (oxford.universitypressscholarship.com). (c) Copyright Oxford University Press, 2021. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in OSO for personal use. date: 21 September 2021

Classical Approximation

Classical Approximation

(p.1) Chapter 1 Classical Approximation
Selected Topics in Approximation and Computation

Marek A. Kowalski

Krzysztof A. Sikorski

Frank Stenger

Oxford University Press

In this chapter we acquaint the reader with the theory of approximation of elements of normed spaces by elements of their finite dimensional subspaces. The theory of best approximation was originated between 1850 and 1860 by Chebyshev. His results and ideas have been extended and complemented in the 20th century by other eminent mathematicians, such as Bernstein, Jackson, and Kolmogorov. Initially, we present the classical theory of best approximation in the setting of normed spaces. Next, we discuss best approximation in unitary (inner product) spaces, and we present several practically important examples. Finally, we give a reasonably complete presentation of best uniform approximation, along with examples, the Remez algorithm, and including converse theorems about best approximation. The goal of this section is to present some general results on approximation in normed spaces.

Keywords:   Abel summation formula, Banach, Chebysheff, Darboux, Fourier series, Gram matrix, Hahn—Banach theorem, Jackson operator, Korovkin operators, Lagrange method

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