Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

For citation:

Pleshakov M. G., Tyshkevich S. V. One counterexample of shape-preserving approximation. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2014, vol. 14, iss. 2, pp. 144-150. DOI: 10.18500/1816-9791-2014-14-2-144-150

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
Full text:
(downloads: 58)

One counterexample of shape-preserving approximation

Pleshakov Mixail Gennadevich, Saratov State University
Tyshkevich Sergey Viktorovich, Saratov State University

Let 2s points yi=−π≤y2s

  1. Jackson D. On approximation by trigonometric sums and polynomials. Trans. Amer. Math. Soc., 1912, vol. 13, pp. 491–515. DOI: http://dx.doi.org/10.1090/S0002-9947-1912-1500930-2.
  2. Zygmund A. Smooth Functions. Duke Math. J., 1945, vol. 12, no. 1, pp. 47–76.
  3. Stechkin S. B. O nailuchshem priblizhenii periodicheskikh funktsii trigonometricheskimi polinomami [On the best approximation of periodic functions by trigonometric polynomials]. Dokl. Akad. Nauk SSSR, 1952, vol. 83, no. 5, pp. 651–654 (in Russian).
  4. Kopotun K. A. Uniform estimates of the coconvex approximation of functions by polynomials. Math. Notes, 1992, vol. 51, no. 3, pp. 245-–254.
  5. Timan A. F. Usilenie teoremy Dzheksona o nailuchshem priblizhenii nepreryvnykh funktsii na konechnom otrezke veshchestvennoi osi [The strengthening of the theorem of Jackson on the best approximation of continuous functions on a finite interval of the real axis]. Dokl. Akad. Nauk SSSR, 1951, vol. 78, no. 1, pp. 17–20 (in Russian).
  6. Dzyadyk V. K. O priblizhenii funktsii obyknovennymimnogochlenami na konechnom otrezke veshchestvennoi osi [On the approximation of functions by ordinary polynomials on a finite interval of the real axis]. Izvestiia AN SSSR. Ser. matematicheskaia, 1958, vol. 22, no. 3, pp. 337–354 (in Russian).
  7. Freud G. ¨Uber die Approximation Reelen Stetiger Functionen Durch Gewohnliche Polinome. Math. Ann., 1959, vol. 137, no. 1, pp. 17–25.
  8. Teljakovskii S. A. Two theorems on approximation of functions by algebraic polynomials. Mat. Sb. (N. S.), 1966, vol. 70(112), no. 2, pp. 252–265 (in Russian).
  9. Brudnyi Yu. A. The approximation of functions by algebraic polynomials. Mathematics of the USSR-Izvestiya, 1968, vol. 2, no. 4, pp. 735–743. DOI: 10.1070/IM1968v002n04ABEH000662
  10. Lorentz G. G., Zeller K. L. Degree of Approximation by Monotone Polynomials. II. J. Approx. Theory, 1969, vol. 2, no. 3, pp. 265–269.
  11. Shevchuk I. A. Priblizhenie mnogochlenami i sledy nepreryvnykh na otrezke funktsii [Approximation by polynomials and traces continuous on the interval functions]. Kiev, Naukova dumka, 1992. 225 p. (in Russian)
  12. Shvedov A. S. Jackson’s theorem in Lp, 0 < p < 1, for algebraic polynomials, and orders of comonotone approximations. Math. Notes, 1979, vol. 25, no. 1, pp. 57–63.
  13. Shvedov A. S. Komonotonnoe priblizhenie funktsii mnogochlenami [Comonotone approximation of functions by polynomials]. Dokl. Akad. Nauk SSSR. 1980, vol. 250, no. 1, pp. 39–42 (in Russian).
  14. Dzyadyk V. K. Vvedenie v teoriiu ravnomernogo priblizheniia funktsii polinomami [Introduction to the theory of uniform approximation of functions by polynomials]. Moscow, Nauka, 1977, 512 p. (in Russian)
Short text (in English):
(downloads: 25)