Abstract
We study how the well-known criterion for the uniform convergence of a sine series with monotone coefficients changes if, instead of monotonicity, one imposes the condition of \(\alpha\)-monotonicity with \(0<\alpha <1\). Moreover, we obtain an addition to the well-known Kolmogorov theorem on the integrability of the sum of a cosine series with convex coefficients tending to zero.
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References
E. D. Alferova and A. Yu. Popov, “On the positivity of average sums of sine series with monotone coefficients,” Math. Notes 110, 623–627 (2021).
A. Yu. Popov, “Refinement of estimates of sums of sine series with monotone coefficients and cosine series with convex coefficients,” Math. Notes 109 (5), 808–818 (2021).
A. Yu. Popov and A. P. Solodov, “Optimal two-sided estimates on the interval \([\pi/2,\pi]\) of the sum of the sine series with convex coefficient sequence,” Math. Notes 112 (2), 328–331 (2022).
T. W. Chaundy and A. E. Jolliffe, “The uniform convergence of a certain class of trigonometric series,” Proc. London Math. Soc. (2) 15, 214–216 (1916).
S. Yu. Tikhonov, “On the uniform convergence of trigonometric series,” Math. Notes 81 (2), 268–274 (2007).
A. S. Belov, M. I. Dyachenko, and S. Yu. Tikhonov, “Functions with general monotone Fourier coefficients,” Russ. Math. Surv. 76 (6), 951–1017 (2021).
A. Zygmund, Trigonometric Series (Cambridge Univ. Press, Cambridge, 1960), Vol. 1.
M. I. Dyachenko, “Trigonometric series with generalized-monotone coefficients,” Sov. Math. (Iz. VUZ) 30 (7), 54–66 (1986).
A. N. Kolmogoroff, “Sur l’ordre de grandeur des coefficients de la serie de Fourier–Lebesgue,” Bull. Acad. Polon., 83–86 (1923).
L. A. Bakashov and S. A. Telyakovskij, “Some properties of lacunary series and the integrability of trigonometric series,” in Analytic Number Theory, Mathematical Analysis, and Their Applications, Proc. Steklov Inst. Math. (1980), Vol. 143, pp. 33–43.
A. F. Andersen, “Comparison theorems in the theory of Cesaro summability,” Proc. London Math. Soc. (2) 27 (1), 39–71 (1927).
E. D. Alferova and M. I. Dyachenko, “\(\alpha\)-monotone sequences and the Lorentz theorem,” Mosc. Univ. Math. Bull. 78 (2), 95–99 (2023).
Funding
This work was carried out at Lomonosov Moscow State University and financially supported by the Russian Science Foundation, project 22-21-00545, https://rscf.ru/en/project/22-21-00545/.
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Translated from Matematicheskie Zametki, 2023, Vol. 114, pp. 339–346 https://doi.org/10.4213/mzm13875.
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Dyachenko, M.I. Uniform Convergence of Sine Series with Fractional-Monotone Coefficients. Math Notes 114, 296–302 (2023). https://doi.org/10.1134/S000143462309002X
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DOI: https://doi.org/10.1134/S000143462309002X