Abstract
It is shown that the exact order of decrease of the norm in L of the remainder of a Fourier sine series with monotone coefficients can be expressed in terms of the coefficients of the series just as for a series with convex coefficients. But the numerical multipliers in the estimates are different.
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References
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R. E. Edwards, Fourier Series: A Modern Introduction (Springer-Verlag, Heidelberg, 1982; Mir, Moscow, 1985), Vol. 1.
S. A. Telyakovskii and G. A. Fomin, “On the convergence in the L-metric of Fourier series with quasi-monotone coefficients,” in Trudy Mat. Inst. Steklov, Vol. 134: Theory of Functions and Its Applications [MIAN, Moscow, 1975], pp. 310–313 [Proc. Steklov Inst. Math. 134, 351–355 (1977)].
S. A. Telyakovskii, “An estimate of the norm of a function by its Fourier coefficients that is suitable in problems of approximation theory,” in Trudy Mat. Inst. Steklov, Vol. 109: Approximation of Periodic Functions [MIAN, Moscow, 1971], pp. 65–97.
S. A. Telyakovskii, “Approximation of differentiable functions by partial sums of their Fourier series,” Mat. Zametki 4 (3), 291–300 (1968).
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Original Russian Text © S. A. Telyakovskii, 2016, published in Matematicheskie Zametki, 2016, Vol. 100, No. 3, pp. 450–454.
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Telyakovskii, S.A. On the rate of convergence in L of Fourier sine series with monotone coefficients. Math Notes 100, 472–476 (2016). https://doi.org/10.1134/S0001434616090145
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DOI: https://doi.org/10.1134/S0001434616090145