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Negative result in pointwise 3-convex polynomial approximation

  • A. V. Bondarenko
  • J. J. Gilewicz
Brief Communications
Let Δ3 be the set of functions three times continuously differentiable on [−1, 1] and such that f″′(x) ≥ 0, x ∈ [−1, 1]. We prove that, for any n ∈ ℕ and r ≥ 5, there exists a function fC r [−1, 1] ⋂ Δ3 [−1, 1] such that ∥f (r) C[−1, 1] ≤ 1 and, for an arbitrary algebraic polynomial P ∈ Δ3 [−1, 1], there exists x such that
$$ \left| {f(x) - P(x)} \right| \geq C\sqrt n {{\uprho}}_n^r(x), $$
where C > 0 is a constant that depends only on r, and
$$ {{{\uprho }}_n}(x): = \frac{1}{{{n^2}}} + \frac{1}{n}\sqrt {1 - {x^2}} . $$

Keywords

Function Convex Naukova Dumka Polynomial Approximation Uniform Approximation Lagrange Form 
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

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  • A. V. Bondarenko
    • 1
  • J. J. Gilewicz
    • 2
  1. 1.Shevchenko Kyiv National UniversityKyivUkraine
  2. 2.Centre de Physique ThéoriqueMarseilleFrance

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