Skip to main content
SpringerLink
Log in

On the Approximation of Functions by Polynomials and Entire Functions of Exponential Type

  • Published:
Ukrainian Mathematical Journal Aims and scope

We present a brief survey of works in the approximation theory of functions known to the author and connected with V. K. Dzyadyk’s research works.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ch. de la Vallée Poussin, Leçons sur l’Approximation des Fonctions d’Une Variable Réelle, Gautier-Villars, Paris (1919).

  2. J. Favard, “Application de la formule sommatoire d’Euler á la démonstration de quelques propriétés extrémales des intégrales des fonctions périodiques ou presque périodiques,” Mat. Tidskrift København, B. H., 4, 81–94 (1936).

    MATH  Google Scholar 

  3. J. Favard, “Sur les meilleurs procédés d’approximation de certaines classes des fonctions par des polynomes trigonométriques,” Bull. Sci. Math., 61, 207–224, 243–256 (1937).

    MATH  Google Scholar 

  4. N. I. Akhiezer and M. G. Krein, “On the best approximation by trigonometric sums of differentiable periodic functions,” Dokl. Akad. Nauk SSSR, 15, 107–111 (1937).

    Google Scholar 

  5. S. N. Bernstein, Collected Works [in Russian], Vol. 2, Akad. Nauk SSSR, Moscow (1954).

  6. A. N. Kolmogorov, “Zur Grössen Ordrung des Restgriedes Fourierischer Reichen differenzierbarer Funktionen,” Ann. Math., 36, 321–326 (1935).

    Google Scholar 

  7. V. K. Dzyadyk, “On the best approximation on a class of periodic functions with bounded sth derivative (0 < s < 1),” Izv. Akad. Nauk SSSR, Ser. Mat., 17, No. 2, 135–162 (1953).

  8. V. K. Dzyadyk, “On the best approximation on a class of periodic functions given by integrals of linear combinations of absolutely monotone kernels,” Mat. Zametki, 16, No. 5, 691–701 (1974).

    MathSciNet  Google Scholar 

  9. R. M. Trigub and E. S. Belinsky, Fourier Analysis and Approximation of Functions, Springer, New York (2004).

    Book  Google Scholar 

  10. N. I. Akhiezer, Lectures on Approximation Theory [in Russian], Nauka, Moscow (1965).

  11. A. F. Timan, Approximation Theory of Functions of Real Variable [in Russian], Fizmatgiz, Moscow (1960).

  12. V. K. Dzyadyk, Introduction to the Theory of Uniform Approximation of Functions by Polynomials [in Russian], Nauka, Moscow (1977).

  13. R. A. de Vore and G. G. Lorentz, Constructive Approximation, Springer, Berlin (1993).

  14. V. K. Dzyadyk, “On the least upper bound of the best approximation of some classes of continuous functions given on the real axis,” Dop. Akad. Nauk Ukr. SSR, Ser. A, 589–592 (1975).

  15. R. M. Trigub, “Exact order of approximation of periodic functions by linear polynomial operators,” East J. Approxim., 15, No. 1, 31–56 (2009).

    MathSciNet  Google Scholar 

  16. R. M. Trigub, “Exact order of approximation of periodic functions by Bernstein–Stechkin polynomials,” Mat. Sb., 204, No. 12, 127–146 (2013).

    Article  MathSciNet  Google Scholar 

  17. R. M. Trigub, “Absolute convergence of Fourier integrals, summability of Fourier series, and approximation of functions by polynomials on a torus,” Izv. Akad. Nauk SSSR, Ser. Mat., 44, No. 6, 1378–1409 (1980).

    MathSciNet  MATH  Google Scholar 

  18. V. K. Dzyadyk and G. A. Alibekov, “On the uniform approximation of functions of complex variable on closed sets with angles,” Mat. Sb., 75, No. 4, 502–557 (1968).

    MathSciNet  Google Scholar 

  19. Yu. A. Brudnyi, “Approximation of functions by algebraic polynomials,” Izv. Akad. Nauk SSSR, Ser. Mat., 32, No. 4, 780–787 (1968).

    MathSciNet  Google Scholar 

  20. R. M. Trigub, “Characteristics of the Lipschitz classes of integer order on a segment on according to the rate of polynomial approximation,” in: Theory of Functions, Functional Analysis, and Their Applications [in Russian], Issue 18 (1973), pp. 63–70.

  21. V. V. Andrievskii, V. I. Belyi, and V. K. Dzyadyk, Conformal Invariants in Constructive Theory of Functions of Complex Variable, World Federation Publ., Atlanta (1995).

  22. R. M. Trigub, “Direct theorems on approximation of smooth functions by algebraic polynomials on a segment,” Mat. Zametki, 54, No. 6, 113–121 (1993).

    MathSciNet  MATH  Google Scholar 

  23. V. P. Motornyi, “Approximation of fractional-order integrals by algebraic polynomials. I,” Ukr. Mat. Zh., 51, No. 5, 603–613 (1999); English translation: Ukr. Math. J., 51, No. 5, 672–683 (1999); V. P. Motornyi, “Approximation of fractional-order integrals by algebraic polynomials. II,” Ukr. Mat. Zh., 51, No. 7, 940–951 (1999); English translation: Ukr. Math. J., 51, No. 7. 1055–1068 (1999).

    Article  MathSciNet  Google Scholar 

  24. A. S. Tovstolis, “Approximation of smooth functions by entire functions with finite semipowers on a semiaxis,” Mat. Zametki, 69, No. 6, 934–943 (2001).

    Article  MathSciNet  Google Scholar 

  25. V. K. Dzyadyk and I. A. Shevchuk, Theory of Uniform Approximation of Functions by Polynomials, de Gruyter, Berlin (2008).

    MATH  Google Scholar 

  26. R. A. de Vore and X. M. Yu, “Pointwise estimates for monotone polynomial approximation,” Constr. Approxim., 1, No. 4, 323–331 (1985).

    MathSciNet  Google Scholar 

  27. R. M. Trigub, “Approximation of the indicator of an interval by algebraic polynomials with Hermitian interpolation at two points,” in: Proc. of the Institute of Applied Mathematics and Mechanics, Ukrainian National Academy of Sciences [in Russian], 4 (1999), pp. 186–194.

  28. R. M. Trigub, “Approximation of functions by polynomials with various constraints,” J. Contemp. Math. Anal., 44, No. 4, 230–242 (2009).

    Article  MathSciNet  Google Scholar 

  29. V. K. Dzyadyk, “Geometric definition of analytic functions,” Usp. Mat. Nauk, 15, No. 1, 191–194 (1960).

    Google Scholar 

  30. V. V. Volchkov, Integral Geometry and Convolution Equations, Springer, New York (2003).

    Book  Google Scholar 

  31. R. M. Trigub, “Approximation of smooth functions and constants by polynomials with integral and natural coefficients,” Mat. Zametki, 70, No. 1, 123–136 (2001).

    Article  MathSciNet  Google Scholar 

  32. Vit. V. Volchkov, “Approximation of analytic functions by polynomials with integer coefficients,” Mat. Zametki, 59, No. 2, 182–186 (1996).

    Article  MathSciNet  Google Scholar 

  33. V. Totik, “Approximation by Bernstein polynomials,” Amer. J. Math., 116, No. 4, 995–1018 (1994).

    Article  MathSciNet  Google Scholar 

  34. S. Foucat, Yu. Kryakin, and A. Shadrin, On the Exact Constant in Jackson–Stechkin Inequality for the Uniform Metric, Preprint arXiv:math/0612283v1[math CA] (2006).

Download references

Author information

Authors and Affiliations

  1. Sumy State University, Sumy, Ukraine

    R. M. Trigub

Authors
  1. R. M. Trigub
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. M. Trigub.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 2, pp. 293–300, February, 2019.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Trigub, R.M. On the Approximation of Functions by Polynomials and Entire Functions of Exponential Type. Ukr Math J 71, 333–341 (2019). https://doi.org/10.1007/s11253-019-01648-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-019-01648-1

Search

Navigation