Ukrainian Mathematical Journal

, Volume 55, Issue 8, pp 1314–1328 | Cite as

Sign-Preserving Approximation of Periodic Functions

  • M. G. Pleshakov
  • P. A. Popov


We prove the Jackson theorem for a zero-preserving approximation of periodic functions (i.e., in the case where the approximating polynomial has the same zeros yi) and for a sign-preserving approximation [i.e., in the case where the approximating polynomial is of the same sign as a function f on each interval (yi, yi − 1)]. Here, yi are the points obtained from the initial points −π ≤ y2sy2s−1 <...< y1 < π using the equality yi = yi + 2s + 2π furthermore, these points are zeros of a 2π-periodic continuous function f.


Continuous Function Initial Point Periodic Function Jackson Theorem 
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.
    D. Leviatan, "Shape-preserving approximation by polynomials," J. Comput. Appl. Math., 121, Nos. 1, 2, 73-94 (2000).Google Scholar
  2. 2.
    S. B. Stechkin, "On the best approximation of periodic functions by trigonometric polynomials," Dokl. Akad. Nauk SSSR, 83, No. 5, 651-654 (1952).Google Scholar
  3. 3.
    V. K. Dzyadyk, Introduction to the Theory of Uniform Approximation of Functions by Polynomials [in Russian], Nauka, Moscow (1977).Google Scholar
  4. 4.
    I. A. Shevchuk, Polynomial Approximation and Traces of Functions Continuous on an Interval [in Russian], Naukova Dumka, Kiev (1992).Google Scholar
  5. 5.
    J. Gilewicz and I. A. Shevchuk, "Comonotone approximation," Fundam. Prikl. Mat., 2, Issue 2, 319-363 (1996).Google Scholar
  6. 6.
    M. G. Pleshakov, "Comonotone Jackson's inequality," J. Approxim. Theory, 99, No. 6, 409-421 (1999).Google Scholar
  7. 7.
    M. G. Pleshakov, Comonotone Approximation of Periodic Functions from Sobolev Classes [in Russian], Author's Abstract of the Candidate-Degree Thesis (Physics and Mathematics), Saratov (1998).Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • M. G. Pleshakov
    • 1
  • P. A. Popov
    • 2
  1. 1.Saratov UniversitySaratovRussia
  2. 2.Kiev National University of Technology and DesignKiev

Personalised recommendations