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Approximation Properties of the de la Vallée-Poussin Method

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Abstract

We present a survey of results concerning the approximation of classes of periodic functions by the de la Vallée-Poussin sums obtained by various authors in the 20th century.

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REFERENCES

  1. C. de la Vallée Poussin, “Sur la meilleure approxiamtion des fonction d'une variable reclle par des expressions d'order donne,” C. R. Acad. Sci. Paris, 166, 799–802 (1918).

    Google Scholar 

  2. L. Fejér, “Untersuchungen über Fouriesche Reihen,” Math. Ann., 58, 501–569 (1904).

    Google Scholar 

  3. A. I. Stepanets, Uniform Approximations by Trigonometric Polynomials [in Russian], Naukova Dumka, Kiev (1981).

    Google Scholar 

  4. A. I. Stepanets, Classification and Approximation of Periodic Functions [in Russian], Naukova Dumka, Kiev (1987).

    Google Scholar 

  5. A.I. Stepanets, Classes of Periodic Functions and Approximation of Their Elements by Fourier Sums [in Russian], Preprint No. 10, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1983).

    Google Scholar 

  6. A. I. Stepanets, “Classification of periodic functions and the rate of convergence of their Fourier series,” Izv. Akad. Nauk SSSR, Ser. Mat., 50, No. 1, 101–136 (1986).

    Google Scholar 

  7. A. I. Stepanets, “Approximation by entire functions in the uniform metric,” in: Approximations by Entire Functions on the Real Axis [in Russian], Preprint No. 27, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1988), pp. 3–41.

    Google Scholar 

  8. A. I. Stepanets, “Approximation of functions given on the real axis by Fourier operators,” Ukr. Mat. Zh., 40, No. 2, 198–209 (1988).

    Google Scholar 

  9. A. I. Stepanets, “Classes of functions given on the real axis and their approximation by entire functions. I,” Ukr. Mat. Zh., 42, No. 1, 102–112 (1990).

    Google Scholar 

  10. A. I. Stepanets, “Classes of functions given on the real axis and their approximation by entire functions. II,” Ukr. Mat. Zh., 42, No. 2, 210–222 (1990).

    Google Scholar 

  11. A. I. Stepanets, “Approximations in spaces of locally integrable functions,” Ukr. Mat. Zh., 46, No. 5, 597–625 (1994).

    Google Scholar 

  12. A. I. Stepanets, Approximation in Spaces of Locally Integrable Functions [in Russian], Preprint No. 18, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1993).

    Google Scholar 

  13. A. I. Stepanets, Approximation of \(\overline \psi \) -Integrals of Periodic Functions by Fourier Sums [in Russian], Preprint No. 11, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1996).

    Google Scholar 

  14. B. Nagy, “Sur une classe générale de procédés de sommation pour les séries de Fourier,” Hung. Acta Math., 1, No. 3, 14–52 (1948).

    Google Scholar 

  15. A. V. Efimov, “On approximation of periodic functions by de la Vallée-Poussin sums. I,” Izv. Akad. Nauk SSSR, Ser. Mat., 23, No. 5, 737–770 (1959).

    Google Scholar 

  16. A. V. Efimov, “On approximation of periodic functions by de la Vallée-Poussin sums. II,” Izv. Akad. Nauk SSSR, Ser. Mat., 24, No. 3, 431–468 (1960).

    Google Scholar 

  17. S. P. Baiborodov, “Approximation of functions by de la Vallée-Poussin sums,” Mat. Zametki, 27, No. 1, 33–48 (1980).

    Google Scholar 

  18. I. P. Natanson, Constructive Theory of Functions [in Russian], Gostekhteoretizdat, Moscow (1949).

    Google Scholar 

  19. S. M. Lozinskii, “On one class of linear operations,” Dokl. Akad. Nauk SSSR, 61, No. 2, 193–196 (1948).

    Google Scholar 

  20. A. F. Timan, “On generalization of some results of A. N. Kolmogorov and S. M. Nikol'skii,” Dokl. Akad. Nauk SSSR, 81, No. 4, 509–511 (1951).

    Google Scholar 

  21. A. F. Timan, “Approximation properties of linear summation methods for Fourier series,” Izv. Akad. Nauk SSSR, Ser. Mat., 17, 99–134 (1953).

    Google Scholar 

  22. A. N. Kulik, “On the remainder of the deviation of de la Vallée-Poussin sums on classes of differentiable functions,” in: Some Problems of the Theory of Approximation of Functions [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1985), pp. 57–62.

    Google Scholar 

  23. A. Kolmogoroff, “Zur Grossenordnung des Restgludes Fourierchen differenzier barer Funktionen,” Ann. Math., 36, 521–526 (1935).

    Google Scholar 

  24. V. T. Pinkevich, “On the order of the remainder of Fourier series for functions differentiable in the Weyl sense,” Izv. Akad. Nauk SSSR, Ser. Mat., 4, No. 5, 521–528 (1940).

    Google Scholar 

  25. S. M. Nikol'skii, “Approximation of periodic functions by trigonometric polynomials,” Tr. Mat. Inst. Akad Nauk SSSR, 15, 3–73 (1945).

    Google Scholar 

  26. S. A. Telyakovskii, “On the norms of trigonometric polynomials and approximation of differentiable functions by linear means of their Fourier series. I,” Tr. Mat. Inst. Akad Nauk SSSR, 62, 61–97 (1961).

    Google Scholar 

  27. S. A. Telyakovskii, “On the approximation of differentiable functions by linear means of their Fourier series,” Izv. Akad. Nauk SSSR, Ser. Mat., 24, No. 2, 213–242 (1960).

    Google Scholar 

  28. S. B. Stechkin, “On de la Vallée-Poussin sums,” Dokl. Akad. Nauk SSSR, 80, No. 4, 545–548 (1951).

    Google Scholar 

  29. S. M. Nikol'skii, “Approximation of functions by trigonometric polynomials in the mean,” Izv. Akad. Nauk SSSR, Ser. Mat., 10, No. 3, 207–256 (1946).

    Google Scholar 

  30. B. Nagy, “Ñber geuisse Extremalfragen bei transformierten trigonometrischen Entwicklungen,” Ber. Acad. Dtsch. Wiss., 90, 103–134 (1938).

    Google Scholar 

  31. N. P. Korneichuk, Extremal Problems in Approximation Theory [in Russian], Nauka, Moscow (1976).

    Google Scholar 

  32. V. K. Dzyadyk and A. I. Stepanets, “On the sequence of zeros of the integral sine,” in: Metric Problems in the Theory of Functions and Mappings [in Russian], Issue 2, Naukova Dumka, Kiev (1971), pp. 64–73.

    Google Scholar 

  33. V. K. Dzyadyk and A. I. Stepanets, “Asymptotic equalities for the least upper bounds for approximations of functions of Hölder classes by Rogosinski polynomials,” Ukr. Mat. Zh., 24, No. 4, 476–487 (1972).

    Google Scholar 

  34. L. A. Ostrovets'kyi, “On asymptotic equalities in the case of approximation of functions of the classes H ω ?by de la Vallée-Poussin sums,” Dopov. Akad. Nauk Ukr. RSR, Ser. A, No. 5, 340–342 (1979).

  35. V. A. Dudas, Approximation of Periodic Functions of Hölder Classes by de la Vallée-Poussin Sums [in Russian], Preprint No. 9, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1980).

    Google Scholar 

  36. V. I. Rukasov, Approximation of Continuous Periodic Functions by Linear Means of Their Fourier Series [in Russian], Preprint No. 62, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1983).

    Google Scholar 

  37. V. I. Rukasov, “Approximation of functions of the class \(C_{\beta, \infty}^\psi\) by linear means of their Fourier series,” Ukr. Mat. Zh., 39, No. 4, 478–483 (1987).

    Google Scholar 

  38. O. A. Novikov and V. I. Rukasov, “Approximation of classes of continuous periodic functions by analogs of de la Vallée-Poussin sums,” in: Fourier Series: Theory and Applications [in Russian], Institute of Mathematics, Ukraninian Academy of Sciences, Kiev (1992), pp. 57–63.

    Google Scholar 

  39. D. N. Bushev, Approximation of Classes of Continuous Periodic Functions by Zygmund Sums [in Russian], Preprint No. 56, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1984).

    Google Scholar 

  40. V. I. Rukasov and O. A. Novikov, “Approximation of analytic functions by de la Vallée-Poussin sums,” in: Fourier Series: Theory and Applications [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1997), pp. 109–110.

    Google Scholar 

  41. A. I. Stepanets and N. L. Pachuliya, On Strong Summability of Fourier Series [in Russian], Preprint No. 61, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1985).

    Google Scholar 

  42. A. N. Kulik, Problems of Approximation of Periodic Differentiable Functions [in Russian], Candidate-Degree Thesis (Physics and Mathematics), Kiev (1987).

  43. O. A. Novikov and V. I. Rukasov, “Approximation of classes of continuous functions by generalized de la Vallée-Poussin sums,” Ukr. Mat. Zh., 47, No. 8, 1069–1079 (1995).

    Google Scholar 

  44. V. I. Rukasov and O.A. Novikov, “Approximation of the classes \(C_{\beta}^\psi\) Hω by generalized de la Vallée-Poussin sums,” Ukr. Mat. Zh., 49, No. 4, 606–610 (1997).

    Google Scholar 

  45. V. I. Rukasov and S. O. Chaichenko, “Approximation of \(\overline \psi \)-integrals of 2π?-periodic functions by de la Vallée-Poussin sums,” in: Fourier Series: Theory and Applications [in Ukrainian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1998), pp. 242–254.

    Google Scholar 

  46. V. I. Rukasov, O. A. Novikov, and S. O. Chaichenko, “Approximation of the classes \(C_{\beta}^{\overline \psi}\) by de la Vallée-Poussin methods,” in: Fourier Series: Theory and Applications [in Ukrainian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (2000), pp. 396–406.

    Google Scholar 

  47. V. I. Rukasov, O. A. Novikov, and S. O. Chaichenko, “Approximation of the classes \(C_{\beta}^{\overline \psi}\) by de la Vallée-Poussin sums (low smoothness),” in: Abstracts of the Ukrainian Mathematical Congress Dedicated to the 200th Anniversary of the Birth of M. V. Ostrohrads'kyi (Kiev, August 21–25, 2001), Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (2001), pp. 51–52.

    Google Scholar 

  48. V. I. Rukasov and S. O. Chaichenko, “Approximation of the classes \(C^{\overline \psi } \) Hω by de la Vallée-Poussin sums,” in: Abstracts of the Ukrainian Mathematical Congress Dedicated to the 200th Anniversary of the Birth of M. V. Ostrohrads'kyi (Kiev, August 21–25, 2001), Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (2001), pp. 52–53.

    Google Scholar 

  49. V. I. Rukasov, “Approximation of functions given on the real axis by de la Vallée-Poussin operators,” Ukr. Mat. Zh., 44, No. 5, 682–690 (1992).

    Google Scholar 

  50. V. A. Baskakov, “ Summation methods for Fourier series that give the best order of approximation on the classes C(ε) ?and their norms,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 8, 60–63 (1981).

  51. V. A. Baskakov, Linear Polynomial Operators with the Best Order of Approximation [in Russian], Kalinin University, Kalinin (1984).

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev

    A. I. Stepanets

  2. Slavyansk Pedagogic Institute, Slavyansk

    V. I. Rukasov

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  1. A. I. Stepanets
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  2. V. I. Rukasov
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Stepanets, A.I., Rukasov, V.I. Approximation Properties of the de la Vallée-Poussin Method. Ukrainian Mathematical Journal 54, 1324–1354 (2002). https://doi.org/10.1023/A:1023435724474

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