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Sharp Two-Sided Estimate for the Sum of a Sine Series with Convex Slowly Varying Sequence of Coefficients

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Abstract

The sum of a sine series \(g\left({b,x} \right) = \sum\nolimits_{k = 1}^\infty {}\)bk sin kx with coefficients forming a convex sequence b is known to be positive on the interval (0,π). To estimate its values near zero Telyakovskiĭ used the piecewise-continuous function \(\sigma \left({{\bf{b}},x} \right) = \left({1/m\left(x \right)} \right)\sum\nolimits_{k = 1}^{m\left(x \right) - 1} {{k^2}\left({{b_k} - {b_{k + 1}}} \right),\,\,m\left(x \right) = \left[{\pi /x} \right]}\). He showed that in some neighborhood of zero the difference g(b,x) − (bm(x)/2) cot(x/2) can be estimated from both sides two-sided in terms of the function σ(b,x) with absolute constants. In the present paper, sharp values of these constants on the class of convex slowly varying sequences b are found. A sharp two-sided estimate for the sum of a sine series on this class is obtained. Examples that demonstrate good accuracy of the obtained two-sided estimates are given.

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References

  1. S. Aljančić, R. Bojanić, and M. Tomić, Sur le comportement asymptotique au voisinage de zéro des séries trigonométriques de sinus à coefficients monotones, Acad. Serbe Sci. Publ. Inst. Math., 10 (1956), 101–120.

    MathSciNet  MATH  Google Scholar 

  2. N. K. Bary, A Treatise on Trigonometric Series, Fizmatgiz (Moscow, 1961) (in Russian); translation, Vols. Pergamon Press Book The Macmillan Co. (New York, 1964).

    Google Scholar 

  3. C.-P. Chen and L. Chen, Asymptotic behavior of trigonometric series with O-regularly varying quasimonotone coefficients, J. Math. Anal. Appl., 250 (2000), 13–26.

    Article  MathSciNet  Google Scholar 

  4. C.-P. Chen and L. Chen, Asymptotic behavior of trigonometric series with O-regularly varying quasimonotone coefficients. II, J. Math. Anal. Appl., 245 (2000), 297–301.

    Article  MathSciNet  Google Scholar 

  5. M. I. Dyachenko and S. Yu. Tikhonov, Trigonometric series with lacunary-monotone coefficients, Eurasian Math. J., 3 (2012), 31–52.

    MathSciNet  MATH  Google Scholar 

  6. M. I. Dyachenko and S. Yu. Tikhonov, Smoothness and asymptotic properties of functions with general monotone Fourier coefficients, J. Fourier Anal. Appl., 24 (2018), 1072–1097.

    Article  MathSciNet  Google Scholar 

  7. J. Karamata, Sur un mode de croissance régulière des fonctions, Mathematica (Cluj), 4 (1930), 38–53.

    MATH  Google Scholar 

  8. V. S. Kozyakin, On the asymptotics of cosine series in several variables with power coefficients, J. Commun. Technol. Electron., 60 (2015), 1441–1444.

    Article  Google Scholar 

  9. Xh. Z. Krasniqi, On the behavior near the origin of doudle sine series with monotone coefficients, Math. Bohemica, 134 (2009), 255–273.

    Article  MathSciNet  Google Scholar 

  10. A. Yu. Popov, Estimates of the sums of sine series with monotone coefficients of certain classes, Mat. Zametki, 74 (2003), 877–888 (in Russian); translation in Math. Notes, 74 (2003), 829–840.

    Article  MathSciNet  Google Scholar 

  11. R. Salem, Détermination de l’ordre de grandeur á l’origine de certaines séries trigonométriques, C. R. Acad. Sci. Paris, 186 (1928), 1804–1806.

    MATH  Google Scholar 

  12. E. Seneta, Regularly Varying Functions, Lecture Notes in Math., vol. 508, Springer-Verlag (Berlin Heidelberg, New York, 1976).

    Book  Google Scholar 

  13. A. P. Solodov, A sharp lower bound for the sum of a sine series with convex coefficients, Mat. Sb., 207 (2016), 124–158 (in Russian); translation in Sb. Math., 207 (2016), 1743–1777.

    Article  Google Scholar 

  14. S. A. Telyakovskiĭ, On the behavior near the origin of the sine series with convex coefficients, Publ. Inst. Math. (Belgrad) Nouvelle série, 58 (1995), 43–50.

    MathSciNet  MATH  Google Scholar 

  15. S. A. Telyakovskiĭ, On the behavior of sine series with convex coefficients near the origin, Dokl. RAN, 357 (1997), 462–463 (in Russian); translation in Dokl. Math., 56 (1997), 913–914.

    MathSciNet  Google Scholar 

  16. S. A. Telyakovskiĭ, On the behavior of sine series near zero, Makedon. Akad. Nauk. Umet. Oddel. Mat.-Tehn. Nauk. Prilozi, 21 (2002), 47–53 (in Russian).

    MathSciNet  Google Scholar 

  17. S. Yu. Tikhonov, Trigonometric series with general monotone coefficients, J. Math. Anal. Appl., 326 (2007), 721–735.

    Article  MathSciNet  Google Scholar 

  18. A. Zygmund, Trigonometric Series, Vols. 3rd ed., Cambridge Univ. Press (Cambridge, 2002).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Lomonosov Moscow State University, 1 Leninskie Gory str., Moscow, 119991, Russia

    A. P. Solodov

  2. Moscow Center for Fundamental and Applied Mathematics, Moscow, Russia

    A. P. Solodov

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  1. A. P. Solodov
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Correspondence to A. P. Solodov.

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This research was supported by RFBR, project number 20-01-00584.

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Solodov, A.P. Sharp Two-Sided Estimate for the Sum of a Sine Series with Convex Slowly Varying Sequence of Coefficients. Anal Math 46, 579–603 (2020). https://doi.org/10.1007/s10476-020-0047-5

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  • DOI: https://doi.org/10.1007/s10476-020-0047-5

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