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Ukrainian Mathematical Journal

, Volume 54, Issue 9, pp 1445–1461 | Cite as

Coconvex Pointwise Approximation

  • G. A. Dzyubenko
  • J. Gilewicz
  • I. A. Shevchuk
Article

Abstract

Assume that a function fC[−1, 1] changes its convexity at a finite collection Y := {y1, ... ys} of s points yi ∈ (−1, 1). For each n > N(Y), we construct an algebraic polynomial Pn of degree ≤ n that is coconvex with f, i.e., it changes its convexity at the same points yi as f and
$$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c{\omega }_{2} \left( {f,\frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in \left[ { - 1,1} \right],$$
where c is an absolute constant, ω2(f, t) is the second modulus of smoothness of f, and if s = 1, then N(Y) = 1. We also give some counterexamples showing that this estimate cannot be extended to the case of higher smoothness.

Keywords

Absolute Constant Finite Collection High Smoothness 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • G. A. Dzyubenko
    • 1
  • J. Gilewicz
    • 2
  • I. A. Shevchuk
    • 3
  1. 1.International Mathematical CenterUkrainian Academy of SciencesKiev
  2. 2.Centre de Physique Théorique CNRSToulon UniversityLuminyFrance
  3. 3.Shevchenko Kiev UniversityKiev

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