A.Yu. Popov, T.V. Rodionov. 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 ... P. 177-190

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

$G_a(x)-\dfrac{x}{2}<g_a(x)<G_a(x)-\dfrac{x}{12}$

and

$F_a(x)+\zeta(a)+\dfrac{\zeta(3)}{4\pi^3}\,x^2\sin(\pi a/2)<f_a(x)<F_a(x)+\zeta(a)+\dfrac{1}{18}\,x^2\sin(\pi a/2)$

are valid for all $a\in(0,1)$ and $x\in(0,\pi]$. 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 $k<1/2$: 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 of the functions $f_a$ and $g_a$ established recently by other authors.

Keywords: special trigonometric series, polylogarithm, periodic zeta function

Received May 19, 2022

Revised July 29, 2022

Accepted August 4, 2022

Funding Agency: The research of the first author (the results of Sections 2–3) was carried out at Moscow State University and supported by the Russian Science Foundation (project no. 22-21-00545). The research of the second author (the results of Section 6) was carried out at Moscow State University and supported by the Russian Foundation for Basic Research (project no. 20-01-00584).

Anton Yur’evich Popov, Dr. Phys.-Math. Sci., Lomonosov Moscow State University and Moscow Centre of Fundamental and Applied Mathematics, Moscow, 119991 Russia, e-mail: station@list.ru

Timofey Victorovich Rodionov, Cand. Sci. (Phys.-Math.), Lomonosov Moscow State University and Moscow Centre of Fundamental and Applied Mathematics, Moscow, 119991 Russia, e-mail: rodionovtv@mail.ru

REFERENCES

1.   Bieberbach L. Analytische Fortsetzung. Berlin: Springer-Verlag, 1955, 168 p. doi: 10.1007/978-3-662-01270-3 . Translated to Russian under the title Analiticheskoe prodolzhenie. Moscow: Nauka Publ., 1967, 240 p.

2.   Zygmund A. Trigonometric series, vol. I, II. Cambridge: Cambridge Univ. Press, 1959; vol. I, 383 p.; vol. II, 354 p. Translated under the title Trigonometricheskie ryady, M.: Mir Publ., 1965, vol. I, 616 p; vol. II, 538 p.

3.   Titchmarsh E.C. The theory of the Riemann zeta-function. Oxford: Oxford Univ. Press, 1987, 422 p. ISBN: 0198533691 . Translated to Russian under the title Teoriya dzeta-funktsii Rimana, Moscow: Izd. Inostr. Lit., 1953, 407 p.

4.   Erdélyi A. (ed.) Higher transcendental functions. Vol. 1. NY: McGraw Hill, 1953, 302 p.

5.   Leau L. Recherches des singularités d’une fonction définie par un développement de Taylor. Journ. de Math. (5), 1899, vol. 5, pp. 365–425.

6.   Liflyand E., Podkorytov A. Lebesgue constants of Riesz type means of negative order. J. Math. Anal. Appl., 2022, vol. 505, no. 2, art. no. 125618. doi: 10.1016/j.jmaa.2021.125618 

7.   Lindelöf E.L. Le calcul des résidus et ses applications à la théorie des fonctions. Paris: Gauthier–Villar, 1905, 158 p.

8.   Olver F.W.J. et al. (eds.) NIST handbook of mathematical functions. NY: Cambridge Univ. Press, 2010, 968 p. The online version: The NIST Digital Library of Mathematical Functions (DLMF): https://dlmf.nist.gov/ 

9.   Truesdell C. On a function which occurs in the theory of the structure of polymers. Ann. Math. (2), 1945, vol. 46, no. 1, pp. 144–157. doi: 10.2307/1969153 

Cite this article as: A.Yu. Popov, T.V. Rodionov. 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. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 4, pp. 177–190; Proceedings of the Steklov Institute of Mathematics (Suppl.) , Vol. 319, Suppl. 1, pp. S204–S217.