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Ukrainian Mathematical Journal

, Volume 71, Issue 2, pp 333–341 | Cite as

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

  • R. M. TrigubEmail author
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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.

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References

  1. 1.
    Ch. de la Vallée Poussin, Leçons sur l’Approximation des Fonctions d’Une Variable Réelle, Gautier-Villars, Paris (1919).Google Scholar
  2. 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).zbMATHGoogle Scholar
  3. 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).zbMATHGoogle Scholar
  4. 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. 5.
    S. N. Bernstein, Collected Works [in Russian], Vol. 2, Akad. Nauk SSSR, Moscow (1954).Google Scholar
  6. 6.
    A. N. Kolmogorov, “Zur Grössen Ordrung des Restgriedes Fourierischer Reichen differenzierbarer Funktionen,” Ann. Math., 36, 321–326 (1935).Google Scholar
  7. 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).Google Scholar
  8. 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).MathSciNetGoogle Scholar
  9. 9.
    R. M. Trigub and E. S. Belinsky, Fourier Analysis and Approximation of Functions, Springer, New York (2004).CrossRefGoogle Scholar
  10. 10.
    N. I. Akhiezer, Lectures on Approximation Theory [in Russian], Nauka, Moscow (1965).Google Scholar
  11. 11.
    A. F. Timan, Approximation Theory of Functions of Real Variable [in Russian], Fizmatgiz, Moscow (1960).Google Scholar
  12. 12.
    V. K. Dzyadyk, Introduction to the Theory of Uniform Approximation of Functions by Polynomials [in Russian], Nauka, Moscow (1977).Google Scholar
  13. 13.
    R. A. de Vore and G. G. Lorentz, Constructive Approximation, Springer, Berlin (1993).Google Scholar
  14. 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).Google Scholar
  15. 15.
    R. M. Trigub, “Exact order of approximation of periodic functions by linear polynomial operators,” East J. Approxim., 15, No. 1, 31–56 (2009).MathSciNetGoogle Scholar
  16. 16.
    R. M. Trigub, “Exact order of approximation of periodic functions by Bernstein–Stechkin polynomials,” Mat. Sb., 204, No. 12, 127–146 (2013).MathSciNetCrossRefGoogle Scholar
  17. 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).MathSciNetzbMATHGoogle Scholar
  18. 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).MathSciNetGoogle Scholar
  19. 19.
    Yu. A. Brudnyi, “Approximation of functions by algebraic polynomials,” Izv. Akad. Nauk SSSR, Ser. Mat., 32, No. 4, 780–787 (1968).MathSciNetGoogle Scholar
  20. 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.Google Scholar
  21. 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).Google Scholar
  22. 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).MathSciNetzbMATHGoogle Scholar
  23. 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).MathSciNetCrossRefGoogle Scholar
  24. 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).MathSciNetCrossRefGoogle Scholar
  25. 25.
    V. K. Dzyadyk and I. A. Shevchuk, Theory of Uniform Approximation of Functions by Polynomials, de Gruyter, Berlin (2008).zbMATHGoogle Scholar
  26. 26.
    R. A. de Vore and X. M. Yu, “Pointwise estimates for monotone polynomial approximation,” Constr. Approxim., 1, No. 4, 323–331 (1985).MathSciNetGoogle Scholar
  27. 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.Google Scholar
  28. 28.
    R. M. Trigub, “Approximation of functions by polynomials with various constraints,” J. Contemp. Math. Anal., 44, No. 4, 230–242 (2009).MathSciNetCrossRefGoogle Scholar
  29. 29.
    V. K. Dzyadyk, “Geometric definition of analytic functions,” Usp. Mat. Nauk, 15, No. 1, 191–194 (1960).Google Scholar
  30. 30.
    V. V. Volchkov, Integral Geometry and Convolution Equations, Springer, New York (2003).CrossRefGoogle Scholar
  31. 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).MathSciNetCrossRefGoogle Scholar
  32. 32.
    Vit. V. Volchkov, “Approximation of analytic functions by polynomials with integer coefficients,” Mat. Zametki, 59, No. 2, 182–186 (1996).MathSciNetCrossRefGoogle Scholar
  33. 33.
    V. Totik, “Approximation by Bernstein polynomials,” Amer. J. Math., 116, No. 4, 995–1018 (1994).MathSciNetCrossRefGoogle Scholar
  34. 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).Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Sumy State UniversitySumyUkraine

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