Literature cited
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).
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).
A. V. Efimov, Linear methods for the summation of Fourier series, Izv. AN SSSR, Ser. matem.,24, No. 5, 743–756 (1960).
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).
S. M. Nikol'skii, Linear methods of summation of Fourier series, Izv. AN SSSR, Ser. matem.,12, No. 3, 259–278 (1948).
J., Karamata and M. Tomich, Summability of Fourier series, Glas Srpske Akad. nauka,206, od. prir.- matem. nauka, 5, 89–126 (1953).
S. B. Stechkin, A problem concerning multipliers for trigonometric polynomials, Dokl. AN SSSR,75, No. 2, 165–168.
A. F. Timan, Notes on trigonometric polynomials and Fourier-Stieltjes series, Uspekhi matem. nauk,12, No. 2, 175–183 (1957).
A. Kolmogoroff, Zur. Grössenordnung des Restgliedes Fourierschen Reihen differenzierbarer Funktionen, Ann. of Math., (2),36, No. 2, 521–526 (1936).
S. M. Nikol'skii, An approximation for periodic functions by trigonometric polynomials, Tr. Matem. in-ta AN SSSR,15 (1945).
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).
A. F. Fiman, Approximating properties of linear methods of summation of Fourier series, Izv. AN SSSR, Ser. matem.,17, No. 2, 99–134 (1953).
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).
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).
S. A. Telyakovskii, Approximations of differentiable functions by de la Vallé Poussin sums, Dokl. AN SSSR,121, No. 3, 426–429 (1958).
S. A. Telyakovskii, Some inequalities for trigonometric series with quasi-convex coefficients, Matem. sb.,63 (105), No. 3, 426–444 (1964).
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).
K. I. Babneko, Approximations for a class of periodic functions by trigonometric polynomials, Dokl. AN SSSR,132, No. 5, 982–985 (1960).
S. A. Telyakovskii, Inequalities for derivatives of trigonometric polynomials in several variables, Sibirskii matem. zh.,4, No. 6, 1404–1411 (1963).
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).
R. P. Boas, Absolute convergence and integrability of trigonometric series, J. Rational Mech. and Anal.,5, No. 4, 621–632 (1956).
L. Tonelli, Serie trigonometriche, Bologna (1928).
G. A. Fomin, Linear methods of summation of Fourier series, Matem. sb.,65 (107), No. 1, 144–152 (1964).
S. A. Telyakovskii, Norms of linear polynomial operators, matem. sb.,68, (110), No. 4, 561–569 (1965).
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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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DOI: https://doi.org/10.1007/BF01094081