Ukrainian Mathematical Journal

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

Coconvex Pointwise Approximation

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


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.


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V. K. Dzyadyk, Introduction to the Theory of Uniform Polynomial Approximation of Functions [in Russian], Nauka, Moscow (1977).Google Scholar
  2. 2.
    S. A. Telyakovskii, “Two theorems on approximation of functions by algebraic polynomials”, Mat. Sb., 70, 252–265 (1966).Google Scholar
  3. 3.
    R. A. DeVore, “Degree of approximation,” in: G. G. Lorentz, C. K. Chui, and L. Schumaker (editors), Approximation Theory. II, Academic Press, New York (1976), pp. 117–162.Google Scholar
  4. 4.
    X. M. Yu, “Pointwise estimates for convex polynomial approximation,” Approxim. Theory Appl., 1, No. 4. 65–74 (1985).Google Scholar
  5. 5.
    H. H. Gonska, D. Leviatan, I. A. Shevchuk, and H.-J. Wenz,“Interpolatory pointwise estimates for polynomial approximation,” Constr. Approxim., 16, 603–629 (2000).Google Scholar
  6. 6.
    D. Leviatan, “Shape-preserving approximation by polynomials,” J. Comp. Appl. Math., 121, 73–94 (2000).Google Scholar
  7. 7.
    G. A. Dzyubenko, J. Gilewicz, and I. A. Shevchuk, “Piecewise monotone pointwise approximation” Constr. Approxim., 141, 311–348 (1998).Google Scholar
  8. 8.
    D. Leviatan, “Pointwise estimates for convex polynomial approximation,” Proc. Amer. Math. Soc., 98, 471–474 (1986).Google Scholar
  9. 9.
    K. A. Kopotun, “Pointwise and uniform estimates for convex approximation of functions by algebraic polynomials,” Constr. Approxim., 10, 153–178 (1994).Google Scholar
  10. 10.
    L. P. Yushchenko, “On one counterexample in convex approximation,” Ukr. Mat. Zh., 52, No. 12, 1715–1721 (2000).Google Scholar
  11. 11.
    X. Wu and S. P. Zhou, “A counterexample in comonotone approximation in L p space,” Colloq. Math., 64, 265–274 (1993).Google Scholar
  12. 12.
    K. A. Kopotun, D. Leviatan, and I. A. Shevchuk, “The degree of coconvex polynomial approximation,” Proc. Amer. Math. Soc., 127, 409–415 (1999).Google Scholar
  13. 13.
    D. Leviatan and I. A. Shevchuk, “Coconvex approximation,” J. Approxim. Theory (to appear).Google Scholar
  14. 14.
    G. A. Dzyubenko and J. Gilewicz, “Nearly coconvex pointwise approximation,” East J. Approxim., 6, 357–383 (2000).Google Scholar
  15. 15.
    J. Gilewicz and L. P. Yushchenko, Piecewise q-Coconvex Pointwise Approximation, Preprint No. CPT-2001, Centre de Physique Théorique CNRS, Luminy, Marseille (2001).Google Scholar
  16. 16.
    I. A. Shevchuk, Polynomial Approximation and Traces of Functions Continuous on a Segment [in Russian], Naukova Dumka, Kyiv (1992).Google Scholar
  17. 17.
    P. M. Trigub, “Approximation of functions by polynomials with integer coefficients,” Izv. Akad. Nauk SSSR, Ser. Mat., 26, 261–280 (1962).Google Scholar

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

Personalised recommendations