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Ukrainian Mathematical Journal

, Volume 42, Issue 5, pp 513–525 | Cite as

Approximation theory and optimization problems

  • N. P. Korneichuk
Article
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Abstract

In the first part of the article (Secs. 1, 2), a short historical survey is given of the development of analyses in the approximation theory of functions, distinguishing the most important stages and the basic papers, which stimulated investigations at each stage. In the second part (Secs. 3, 4), the basic aspects of the contemporary state of approximation theory and some tendencies of its further development are illuminated, and new statements of problems are formulated, connected with the optimization of approximation methods.

Keywords

Approximation Method Approximation Theory Basic Aspect Important Stage Historical Survey 
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.

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Literature cited

  1. 1.
    P. L. Chebyshev, “Questions regarding the least magnitudes connected with the approximate representation of functions,” Zap. Akad. Nauk (1859) [Collected Works, pp. 462–578, Izd. Akad. Nauk SSSR, Moscow (1955)].Google Scholar
  2. 2.
    K. Weierstrass, “Über die analytische Darstellung sogenannter willkiirlicher Funktionen einerieelen Veränderlichen,” Sitzungber. Acad. Berlin, 633–639, 789–805 (1885).Google Scholar
  3. 3.
    D. Jackson, Über die Genauigkeit des Annäherungstetigen Funktionen durch ganze rationale Funktionen gegebenen Graden und trigonometrischen Summen gegebener Ordnung, Dissertation, Gottingen (1911).Google Scholar
  4. 4.
    S. N. Bernshtein, “The best approximation of continuous functions by means of polynomials of a given degree,” Soobshch. Khar'k. Mat. Obshch., Ser. 2,13, 49–194 (1912).Google Scholar
  5. 5.
    J. La Vallée-Poussin, Ch. Lecons sur l'Approximation des Fonctions d'une Variable Réelle, Gautier-Villars, Paris (1919).Google Scholar
  6. 6.
    A. Zygmund, “Smooth functions,” Duke Math. J.,12, No. 1, 47–76 (1945).Google Scholar
  7. 7.
    S. M. Nikol'skii, “The best approximation by polynomials of functions satisfying a Lipschitz condition,” Izv. Akad. Nauk SSSR, Ser. Mat.,10, No. 4, 295–322 (1946).Google Scholar
  8. 8.
    S. B. Stechkin, “The order of best approximations of continuous functions,” Izv. Akad. Nauk SSSR, Ser. Mat.,15, No. 3, 219–242 (1951).Google Scholar
  9. 9.
    V. K. Dzyadyk, “The constructive nature of functions satisfying the condition Lip α (0 < α < 1) on a finite segment of the real axis,” Izv. Akad. Nauk SSSR, Ser. Mat.,20, No. 2, 623–642 (1956).Google Scholar
  10. 10.
    H. Lebesgue, “Sur les integrales singulièrs,” Ann. de Toulouse,1, 25–117 (1909).Google Scholar
  11. 11.
    A. N. Kolmogoroff, “Zur Grossenordnung des restgliedes Fourierscher Reihen differenzierbarer Funktionen,” Ann. Math.,36, 521–526 (1935).Google Scholar
  12. 12.
    S. M. Nikol'skii, “The approximation of periodic functions by trigonometric polynomials,” Tr. Mat. Inst. Akad. Nauk SSSR,15, 1–76 (1945).Google Scholar
  13. 13.
    L. Fejér, “Untersuchungen über Fouriersche Reihen,” Math. Ann.,58, 501–569 (1904).Google Scholar
  14. 14.
    J. Favard, “Sur l'approximation des fonctions périodiques par des polynômes trigonométriques,” C. R. Acad. Sci.,203, 1122–1124 (1936).Google Scholar
  15. 15.
    J. Favard, “Sur les meilleurs procèdes d'approximation des certaines classes des fonctions par des polynômes trigonometriques,” Bull. Sci. Math., Ser. 2,60, 209–224, 243–256 (1937).Google Scholar
  16. 16.
    A. N. Kolmogoroff, “Über die besste Annährung von Funktionen einer gegebenen Funktion-klassen,” Ann. Math.,37, 107–110 (1936).Google Scholar
  17. 17.
    A. N. Kolmogorov, “Inequalities between the upper bounds of sequential derivatives of functions on an infinite interval,” Uch. Zap. Mosk. Univ. Matematika,30, No. 3, 3–13 (1939).Google Scholar
  18. 18.
    B. Sz. Nagy, “Über gewisse Extremalfragen bei transformierten trigonometrischen Entwicklungen. 1. Periodischer Fall,” Ber. Math.-Phys. Acad. d. Wiss.,90, 103–134 (1938).Google Scholar
  19. 19.
    S. M. Nikol'skii, “The approximation of functions by trigonometric polynomials in the mean,” Izv. Akad. Nauk SSSR, Ser. Mat.,10, No. 3, 207–256 (1946).Google Scholar
  20. 20.
    N. P. Korneichuk, “Extremal values of functionals and the best approximation on classes of periodic functions,” Izv. Akad. Nauk SSSR, Ser. Mat.,35, No. 1, 93–124 (1971).Google Scholar
  21. 21.
    K. Borsuk, “Drei Sätze über die n-dimensionale euklidische Späre,” Fund. Math.,20, 177–191 (1933).Google Scholar
  22. 22.
    V. M. Tikhomirov, “The diameters of sets in function spaces and the theory of best approximations,” Usp. Mat. Nauk,15, No. 3, 81–120 (1960).Google Scholar
  23. 23.
    N. P. Korneichuk, Splines in Approximation Theory [in Russian], Nauka, Moscow (1984).Google Scholar
  24. 24.
    S. M. Nikol'skii, “The question of evaluating approximations by quadrature formulas,” Usp. Mat. Nauk,5, No. 2 (36), 165–177 (1950).Google Scholar
  25. 25.
    S. M. Nikol'skii, Quadrature Formulas [in Russian], Fizmatgiz, Moscow (1958).Google Scholar
  26. 26.
    S. M. Nikol'skii, The Approximation of Functions of Many Variables and Imbedding Theorems [in Russian], Nauka, Moscow (1977).Google Scholar
  27. 27.
    K. I. Babenko, “The approximation of periodic functions of many variables by trigonometric polynomials,” Dokl. Akad. Nauk SSSR,132, No. 2, 247–250 (1960).Google Scholar
  28. 28.
    V. N. Temlyakov, “The approximation of functions with a bounded mixed derivative,” Tr. Mat. Inst. Akad. Nauk SSSR,178, 1–112 (1986).Google Scholar
  29. 29.
    S. M. Nikol'skii and P. I. Lizorkin, “Approximation by spherical polynomials,” Tr. Mat. Inst. Akad. Nauk SSSR,166, 186–200 (1984).Google Scholar
  30. 30.
    V. N. Konovalov, “An approximation theorem of Jackson type for functions of many variables,” Ukr. Mat. Zh.,33, No. 6, 757–764 (1981).Google Scholar
  31. 31.
    B. Sendov, Hausdorff Approximations [in Russian], Bulgarian Academy of Sciences, Sofia (199).Google Scholar
  32. 32.
    Bl. Sendov and V. A. Popov, The Averaged Moduli of Smoothness: Applications in Numerical Methods and Approximation, Wiley, New York (1988).Google Scholar
  33. 33.
    V. K. Dzyadyk, Introduction to the Theory of Uniform Approximation of Functions by Polynomials [in Russian], Nauka, Moscow (1977).Google Scholar
  34. 34.
    A. I. Stepanets, Uniform Approximations by Trigonometric Polynomials [in Russian], Naukova Dumka, Kiev (1981).Google Scholar
  35. 35.
    A. I. Stepanets, Classification and the Approximation of Periodic Functions [in Russian], Naukova Dumka, Kiev (1987).Google Scholar
  36. 36.
    J. H. Ahlberg, E. N. Nilson, and J. L. Walsh, The Theory of Splines and Their Applications, Academic Press, New York (1967).Google Scholar
  37. 37.
    S. B. Stechkin and Yu. N. Subbotin, Splines in Numerical Mathematics [in Russian], Nauka, Moscow (1976).Google Scholar
  38. 38.
    N. P. Korneichuk, Exact Constants in Approximation Theory [in Russian], Nauka, Moscow (1987).Google Scholar
  39. 39.
    Yu. S. Zav'yalov, B. I. Kvasov, and V. L. Miroshnichenko, Methods of Spline-Functions [in Russian], Nauka, Moscow (1980).Google Scholar
  40. 40.
    V. F. Babenko, “Asymptotic extremal problems of approximation theory,” Dokl. Akad. Nauk SSSR,269, No. 3, 521–524 (1983).Google Scholar
  41. 41.
    Yu. S. Zav'yalov, V. A. Leus, and V. A. Skorospelov, Splines in Engineering Geometry [in Russian], Mashinostroenie, Moscow (1985).Google Scholar
  42. 42.
    P. M. Prenter, Splines and Variational Methods, Wiley, New York (1975).Google Scholar
  43. 43.
    N. P. Korneichuk, “New results on extremal problems of the theory of quadrature,” in: Quadrature Formulas [in Russian], S. M. Nikol'skii (ed.), 4th ed., Nauka, Moscow (1988).Google Scholar
  44. 44.
    A. A. Zhensykbaev, “Monosplines of minimal norm and the best quadrature formulas,” Usp. Mat. Nauk.,36, No. 4 (220), 107–159 (1981).Google Scholar
  45. 45.
    V. N. Temlyakov, “The approximate recovery of periodic functions of several variables,” Mat. Sb.,128, No. 2, 256–268 (1985).Google Scholar
  46. 46.
    B. S. Kashin, “The diameters of some finite-dimensional sets and classes of smooth functions,” Izv. Akad. Nauk SSSR, Ser. Mat.,41, No. 2, 234–251 (1977).Google Scholar
  47. 47.
    V. M. Tikhomirov, “Approximation theory,” Vol. 14, pp. 103–260, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz Inst. Nauchn. i Tekh. Inform., Moscow (1987).Google Scholar
  48. 48.
    V. M. Tikhomirov, Some Problems of Approximation Theory [in Russian], Moscow State University (1976).Google Scholar
  49. 49.
    A. Pinkus, n-Widths in Approximation Theory, Springer, Berlin (1985).Google Scholar
  50. 50.
    Sh. M. Galeev, “Kolmogorovian diameters of classes of periodic functions of many variables in the space Lq,” Izv. Akad. Nauk SSSR, Ser. Mat.,49, No. 5, 916–934 (1985).Google Scholar
  51. 51.
    N. P. Korneichuk, “The optimal coding of elements of a metric space,” Ukr. Mat. Zh.,39, No. 2, 168–173 (1987).Google Scholar
  52. 52.
    N. P. Korneichuk, “Approximation and optimal coding of smooth planar curves,” Ukr. Mat. Zh.,41, No. 4, 492–499 (1989).Google Scholar
  53. 53.
    J. F. Traub and H. Wozniakowski, A General Theory of Optimal Algorithms, Academic Press (1980).Google Scholar
  54. 54.
    S. V. Pereverzev, “The complexity of the problem of finding solutions of a Fredholm equation of type II with smooth kernels,” Ukr. Mat. Zh.,40, No. 1, 84–91 (part I) (1988);41, No. 2, 189–193 (part II) (1989).Google Scholar

Copyright information

© Plenum Publishing Corporation 1991

Authors and Affiliations

  • N. P. Korneichuk
    • 1
  1. 1.Mathematics InstituteAcademy of Sciences of the Ukrainian SSRKiev

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