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Estimates of sums of trigonometric series and polynomials useful in connection with problems in the theory of approximation of functions

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Literature cited

  1. S. A. Teyakovskii, Norms of trigonometric polynomials and the approximation of differentiable functions by linear means of their Fourier series. I, Tr. Matem., in-ta AN SSSR,62, 61–97 (1961).

    Google Scholar 

  2. S. A. Telyakovskii, Norms of trigonometric polynomials and the approximation of differentiable functions by linear means of their Fourier series, II, Izv. AN SSSR, Ser. matem.,27, No. 2, 253–272 (1963).

    Google Scholar 

  3. A. V. Efimov, Linear methods for the summation of Fourier series, Izv. AN SSSR, Ser. matem.,24, No. 5, 743–756 (1960).

    Google Scholar 

  4. A. V. Efimov, An inequality for the integral of the absolute value of a polynomial on the unit circle. Uspekhi matem. nauk,15, 215–218 (1960).

    Google Scholar 

  5. S. M. Nikol'skii, Linear methods of summation of Fourier series, Izv. AN SSSR, Ser. matem.,12, No. 3, 259–278 (1948).

    Google Scholar 

  6. J., Karamata and M. Tomich, Summability of Fourier series, Glas Srpske Akad. nauka,206, od. prir.- matem. nauka, 5, 89–126 (1953).

  7. S. B. Stechkin, A problem concerning multipliers for trigonometric polynomials, Dokl. AN SSSR,75, No. 2, 165–168.

  8. A. F. Timan, Notes on trigonometric polynomials and Fourier-Stieltjes series, Uspekhi matem. nauk,12, No. 2, 175–183 (1957).

    Google Scholar 

  9. A. Kolmogoroff, Zur. Grössenordnung des Restgliedes Fourierschen Reihen differenzierbarer Funktionen, Ann. of Math., (2),36, No. 2, 521–526 (1936).

    Google Scholar 

  10. S. M. Nikol'skii, An approximation for periodic functions by trigonometric polynomials, Tr. Matem. in-ta AN SSSR,15 (1945).

  11. B. Sz. Nagy, Sur une class générale de procédés de summation pour les séries de Fourier, Hungarica Acta Math.,1, No. 3, 14–52 (1948).

    Google Scholar 

  12. A. F. Fiman, Approximating properties of linear methods of summation of Fourier series, Izv. AN SSSR, Ser. matem.,17, No. 2, 99–134 (1953).

    Google Scholar 

  13. A. V. Efimov, Linear methods for the approximation of certain classes of continuous periodic functions, Tr. Matem. in-ta AN SSSR,62, 3–47 (1961).

    Google Scholar 

  14. S. M. Nikol'skii, The asymptotic properties of the remainder in the approximation of functions satisfying the Lipschitz condition by Féjér sums, Izv. AN SSSR, Ser. Matem.,4, No. 6, 501–508 (1940).

    Google Scholar 

  15. S. A. Telyakovskii, Approximations of differentiable functions by de la Vallé Poussin sums, Dokl. AN SSSR,121, No. 3, 426–429 (1958).

    Google Scholar 

  16. S. A. Telyakovskii, Some inequalities for trigonometric series with quasi-convex coefficients, Matem. sb.,63 (105), No. 3, 426–444 (1964).

    Google Scholar 

  17. A. Kolmogoroff, Sur l'ordre de grandeur des coefficients de la serie de Fourier-Lebesgne. Bull. Acad. Polon., Ser. A., Sci. math., 83–86 (1923).

  18. K. I. Babneko, Approximations for a class of periodic functions by trigonometric polynomials, Dokl. AN SSSR,132, No. 5, 982–985 (1960).

    Google Scholar 

  19. S. A. Telyakovskii, Inequalities for derivatives of trigonometric polynomials in several variables, Sibirskii matem. zh.,4, No. 6, 1404–1411 (1963).

    Google Scholar 

  20. S. A. Telyakovskii, Conditions for the integrability of trigonometric series with applications to the investigation of linear methods of summation of Fourier series, Izv. AN SSSR, Ser. matem.,28, No. 6, 1209–1236 (1964).

    Google Scholar 

  21. R. P. Boas, Absolute convergence and integrability of trigonometric series, J. Rational Mech. and Anal.,5, No. 4, 621–632 (1956).

    Google Scholar 

  22. L. Tonelli, Serie trigonometriche, Bologna (1928).

  23. G. A. Fomin, Linear methods of summation of Fourier series, Matem. sb.,65 (107), No. 1, 144–152 (1964).

    Google Scholar 

  24. S. A. Telyakovskii, Norms of linear polynomial operators, matem. sb.,68, (110), No. 4, 561–569 (1965).

    Google Scholar 

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Translated from Matematicheskie Zametki, Vol. 1, No. 5, pp 611–623, May, 1967. Dissertation defended February 23, 1967.

The author wishes to thank sincerely S. M. Nikol'skii and S. B. Stechkin for their constant interest in his work and their many discussions which were always extremely useful.

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Telyakovskii, S.A. Estimates of sums of trigonometric series and polynomials useful in connection with problems in the theory of approximation of functions. Mathematical Notes of the Academy of Sciences of the USSR 1, 404–412 (1967). https://doi.org/10.1007/BF01094081

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