On the Sum of a Trigonometric Sine Series with Monotone Coefficients

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.