On the Sum of a Trigonometric Sine Series with Monotone Coefficients

A. S. Belov

Informatics and Automation, Approximation Theory, Functional Analysis, and Applications, Tome 319 (2022), pp. 29-50.

Résumé

We prove that for each positive integer $n$ the conjugate Dirichlet kernel ${\tilde{D}}_n(x)=\sum _{k=1}^n\sin \left(kx\right)$ is semiadditive on the interval $[0,2\pi ]$, that is, ${\tilde{D}}_n(x_1)+{\tilde{D}}_n(x_2)\ge {\tilde{D}}_n(x_1+x_2)$ 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 ${\tilde{D}}_n(x_1)={\tilde{D}}_n(x_2)={\tilde{D}}_n(x_1+x_2)=0$ 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.

Détail

Mots-clés : conjugate Dirichlet kernel, semiadditive functions, nonnegative trigonometric polynomials.

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