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Linear k-Monotonicity Preserving Algorithms and Their Approximation Properties

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Mathematical Aspects of Computer and Information Sciences (MACIS 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9582))

Abstract

This paper examines the problem of finding the linear algorithm (operator) of finite rank n (i.e. with a n-dimensional range) which gives the minimal error of approximation of identity operator on some set over all finite rank n linear operators preserving the cone of k-monotonicity functions. We introduce the notion of linear relative (shape-preserving) n-width and find asymptotic estimates of linear relative n-widths for linear operators preserving k-monotonicity in the space \(C^k[0,1]\). The estimates show that if linear operator with finite rank n preserves k-monotonicity, the degree of simultaneous approximation of derivative of order \(0\le i\le k\) of continuous functions by derivatives of this operator cannot be better than \(n^{-2}\) even on the set of algebraic polynomials of degree \(k+2\) (as well as on bounded subsets of Sobolev space \(W^{(k+2)}_\infty [0,1]\)).

The results were obtained within the framework of the state task of Russian Ministry of Education and Science (project 1.1520.2014K).

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Authors and Affiliations

  1. Department of Mechanics and Mathematics, Saratov State University, Saratov, Russian Federation

    S. P. Sidorov

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  1. S. P. Sidorov
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Correspondence to S. P. Sidorov .

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Editors and Affiliations

  1. Wilfrid Laurier University, Waterloo, Ontario, Canada

    Ilias S. Kotsireas

  2. Hamburg University of Technology, Hamburg, Germany

    Siegfried M. Rump

  3. New York University, New York, New York, USA

    Chee K. Yap

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Sidorov, S.P. (2016). Linear k-Monotonicity Preserving Algorithms and Their Approximation Properties. In: Kotsireas, I., Rump, S., Yap, C. (eds) Mathematical Aspects of Computer and Information Sciences. MACIS 2015. Lecture Notes in Computer Science(), vol 9582. Springer, Cham. https://doi.org/10.1007/978-3-319-32859-1_7

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  • DOI: https://doi.org/10.1007/978-3-319-32859-1_7

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-32858-4

  • Online ISBN: 978-3-319-32859-1

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