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

Bibliographie

[1] Biberbakh L., Analiticheskoe prodolzhenie, Nauka, M., 1967, 240 pp.

[2] Zigmund A., Trigonometricheskie ryady, v 2 t., v. 1, Mir, M., 1965, 616 pp. ; т. 2, 538 с. | MR

[3] Titchmarsh E.K., Teoriya dzeta-funktsii Rimana, Izd-vo inostrannoi lit-ry, M., 1953, 407 pp. | MR

[4] Higher transcendental functions, v. 1, ed. Erdélyi A., McGraw Hill, NY, 1953, 302 pp.

[5] Leau L., “Recherches des singularités d'une fonction définie par un développement de Taylor”, J. de Math.(5), 5 (1899), 365–425 | MR

[6] Liflyand E., Podkorytov A., “Lebesgue constants of Riesz type means of negative order”, J. Math. Anal. Appl., 505:2 (2022), 125618 | DOI | MR

[7] Lindelöf E.L., Le calcul des résidus et ses applications à la théorie des fonctions, Gauthier-Villar, Paris, 1905, 158 pp. | MR

[8] NIST Handbook of Mathematical Functions, eds. Olver F.W.J. et al., Cambridge Univ. Press, NY, 2010, 968 pp.; The online version: The NIST Digital Library of Mathematical Functions (DLMF) URL: https://dlmf.nist.gov/

[9] Truesdell C., “On a function which occurs in the theory of the structure of polymers”, Ann. Math. (2), 46:1 (1945), 144–157 | DOI | MR