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
$$ {{{\uprho }}_n}(x): = \frac{1}{{{n^2}}} + \frac{1}{n}\sqrt {1 - {x^2}} . $$


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.


  1. 1.
    I. A. Shevchuk, Polynomial Approximation and Traces of Functions Continuous on an Interval [in Russian], Naukova Dumka, Kiev (1992).Google Scholar
  2. 2.
    V. N. Konovalov and D. Leviatan, “Shape-preserving widths of Sobolev-type classes of s -monotone functions on a finite interval,” Isr. J. Math., 133, 239–268 (2003).zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    J. Cao and H. Gonska, “Pointwise estimates for higher-order convexity preserving polynomial approximation,” J. Austr. Math. Soc. Ser. B, 36, 213–233 (1994).zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    A. V. Bondarenko and A. V. Primak, “Negative results in shape-preserving approximation of higher order,” Mat. Zametki, 76, No. 6, 812–823 (2004).MathSciNetGoogle Scholar
  5. 5.
    V. K. Dzyadyk and I. A. Shevchuk, Theory of Uniform Approximation of Functions by Polynomials, de Gruyter, Berlin (2008).zbMATHGoogle Scholar

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

Personalised recommendations