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Linear k-Monotonicity Preserving Algorithms and Their Approximation Properties

  • S. P. SidorovEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, 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 \(W^{(k+2)}_\infty [0,1]\)).

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Mechanics and MathematicsSaratov State UniversitySaratovRussian Federation

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