Geodesic
On the Sum of a Trigonometric Sine Series with Monotone Coefficients
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Approximation Theory, Functional Analysis, and Applications, Tome 319 (2022), pp. 29-50.

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We prove that for each positive integer $n$ the conjugate Dirichlet kernel $\widetilde {D}_n(x)=\sum _{k=1}^{n}\sin (kx)$ is semiadditive on the interval $[0,2\pi ]$, that is, $\widetilde {D}_n(x_1) + \widetilde {D}_n(x_2) \ge \widetilde {D}_n(x_1 + x_2)$ for any nonnegative real numbers $x_1$ and $x_2$ such that $x_1 + x_2\le 2\pi $; moreover, for positive $x_1$ and $x_2$ with $x_1 + x_2 2\pi $, the equality is attained if and only if the condition $\widetilde {D}_n(x_1) = \widetilde {D}_n(x_2) = \widetilde {D}_n(x_1 + x_2) = 0$ is satisfied. We use this property of the conjugate Dirichlet kernel to study the sum of a sine series with monotone coefficients. We also examine the properties of some nonnegative trigonometric polynomials.
Keywords: conjugate Dirichlet kernel, semiadditive functions, nonnegative trigonometric polynomials. 
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A. S. Belov. On the Sum of a Trigonometric Sine Series with Monotone Coefficients. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Approximation Theory, Functional Analysis, and Applications, Tome 319 (2022), pp. 29-50. http://geodesic.mathdoc.fr/item/TM_2022_319_a2/

[1] A. S. Belov, “Power series and Peano curves”, Math. USSR, Izv., 27:1 (1986), 1–26 | DOI | MR

[2] A. S. Belov, “Zeros of the sum of a trigonometric series with monotone coefficients”, Sov. Math., 33:12 (1989), 1–6 | MR

[3] Dyachenko M.I., “O ryadakh Fure s monotonno ubyvayuschimi koeffitsientami i nekotorykh voprosakh gladkosti sopryazhennykh funktsii”, Soobsch. AN Gruz. SSR, 104:3 (1981), 533–536

[4] D'yachenko M.I., “On some properties of trigonometric series with monotone decreasing coefficients”, Anal. math., 10:3 (1984), 193–205 | DOI | MR

[5] Dyachenko M.I., “O raskhodyaschikhsya ryadakh po slabo multiplikativnym sistemam funktsii”, Teoriya funktsii i priblizhenii: Tr. 2-i Sarat. zimn. shk. (24 yanv.–5 fevr. 1984 g.), Ch. 2, Izd-vo Sarat. un-ta, Saratov, 1986, 101–104

[6] Hartman P., Wintner A., “On sine series with monotone coefficients”, J. London Math. Soc., 28:1 (1953), 102–104 | DOI | MR

[7] K. A. Oganesyan, “The measure of the set of zeros of the sum of a nondegenerate sine series with monotone coefficients in the closed interval $[0,\pi ]$”, Math. Notes, 103:3–4 (2018), 621–625 | DOI | MR

[8] K. A. Oganesyan, “A functional inequality for sine series”, Math. Notes, 107:3–4 (2020), 531–533 | DOI | MR