Abstract
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.
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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2022, Vol. 319, pp. 29–50 https://doi.org/10.4213/tm4253.
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Belov, A.S. On the Sum of a Trigonometric Sine Series with Monotone Coefficients. Proc. Steklov Inst. Math. 319, 22–42 (2022). https://doi.org/10.1134/S0081543822050030
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DOI: https://doi.org/10.1134/S0081543822050030