Uniform with Respect to the Parameter $a\in(0,1)$ Two-Sided Estimates of the Sums of Sine and Cosine Series with Coefficients $1/k^a$ by the First Terms of Their Asymptotics

A. Yu. Popov ; T. V. Rodionov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 28 (2022) no. 4, pp. 177-190

Résumé

Uniform with respect to the parameter $a\in(0,1)$ estimates of the functions $f_a(x)=\sum_{k=1}^{\infty}k^{-a}\cos kx$ and $g_a(x)=\sum_{k=1}^{\infty}k^{-a}\sin kx$ by the first terms of their asymptotic expansions $F_a(x)=\sin(\pi a/2)\Gamma(1-a)x^{a-1}$ and $G_a(x)=\cos(\pi a/2)\Gamma(1-a)x^{a-1}$ are obtained. Namely, it is proved that the inequalities $$G_a(x)-\dfrac{x}{2}(x)(x)-\dfrac{x}{12},$$ $$F_a(x)+\zeta(a)+\dfrac{\zeta(3)}{4\pi^3}\,x^2\sin(\pi a/2)(x)(x)+\zeta(a)+\dfrac{1}{18}\,x^2\sin(\pi a/2)$$ are valid for all $a\in(0,1)$ and $x\in(0,\pi]$. \indent It is shown that the estimates are unimprovable in the following sense. In the lower estimate for the sine series, the subtrahend $x/2$ cannot be replaced by $kx$ with any $k1/2$: the estimate ceases to be fulfilled for sufficiently small $x$ and the values of $a$ close to $1$. In the upper estimate, the subtrahend $x/12$ cannot be replaced by $kx$ with any $k>1/12$: the estimate ceases to be fulfilled for the values of $a$ and $x$ close to $0$. In the lower estimate for the cosine series, the multiplier $\zeta(3)/(4\pi^3)$ of $x^2\sin(\pi a/2)$ cannot be replaced by any larger number: the estimate ceases to be fulfilled for $x$ and $a$ close to $0$. In the upper estimate for the cosine series, the multiplier $1/18$ of $x^2\sin(\pi a/2)$ can probably be replaced by a smaller number but not by $1/24$: for every $a\in[0.98,1)$, such an estimate would not hold at the point $x=\pi$ as well as on a certain closed interval $x_0(a)\le x\le\pi$, where $x_0(a)\to0$ as $a\to1-$. The obtained results allow us to refine the estimates for the functions $f_a$ and $g_a$ established recently by other authors.

Détail

Keywords: special trigonometric series, periodic zeta function.
Mots-clés : polylogarithm