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 Chebyshevskii Sb., 2020, Volume 21, Issue 3, Pages 18–28 (Mi cheb924)

On the trigonometric sum modulo subdivision of the real axis

A. A. Artemov, V. N. Chubarikov

Lomonosov Moscow State University (Moscow)

Abstract: The estimate of the trigonometric sum of the kind
$$S=\sum_{a<t_s\leq b}e^{2\pi if(t_s)},$$
where $a\geq 0,a\leq b$ are real numbers, $t_s$ is increasing to infinity of non-negative numbers, $f(t)$ is a smooth real function, is found.
Here also there are proved the analogues of Euler's, Sonin's, Poisson's and van der Corput's formulas for considering sum.
Let be given the sequence of $\Delta$ points
$$0=t_0<t_1<t_2<…<t_s<…, \lim\limits_{n\to\infty}t_n=+\infty,$$
on the positive half-axis of the real line.
For non-negative number $x$ we define the analogue of the integer part $[x]_{\Delta},$ meeting to the sequence $\Delta: [x]_{\Delta}=t_s,$ if $t_s\leq x<t_{s+1}, s\geq 0.$ The fractional part $\{x\}_{\Delta}$ is defined by the equality
$$\{x\}_{\Delta}=\frac{x-t_s}{t_{s+1}-t_s},$$
if $t_s\leq x<t_{s+1}, s\geq 0,$ moreover $0\leq\{x\}_{\Delta}<1.$
We define the analogue of the Bernoulli function meeting to the sequence $\Delta: \rho_\Delta(x)=0,5-\{x\}_\Delta.$
Then is valid the following analogue of the van der Corput's theorem for subdivisions.
Let $\Delta=\{t_s\}, 0=t_0<t_1<…<t_s<…,$ be a subdivision of the half-axis $t\geq 0$ of the real line, $\delta_s=t_{s+1}-t_s\geq 1, \delta(a,b)=\max\limits_{a\leq x\leq b}{\rho'_{\Delta}(x)},$ and let be given the sequence $\Delta_0=\{\mu_s\}, \mu_s=0,5(t_s+t_{s+1}), s\geq 0,$ and points $a,b\in\Delta_0,$ let, also, $f'(x)$ be continuous, monotonic sign-constant in the interval $a< x\leq b,$ moreover there exists the constant $\delta$ such that $0<2\delta\delta^{-1}(a,b)<1$ and that for all $x$ from this interval is valid inequality $|f'(x)|\leq\delta.$ Then we have
$$\sum_{a<t_s\leq b}e^{2\pi if(t_s)}=\int\limits_{a}^{b}\rho'_\Delta(x)e^{2\pi if(x)} dx+10\theta\frac{\delta}{1-\delta\delta^{-1}(a,b)}, |\theta|\leq 1.$$

Keywords: subdivision of the real axis, the trigonometric sum modulo subdivision, Van der Corput's theorem on replacing a trigonometric sum modulo subdivision to an integral, the Euler's, Sonin's, Poisson's summation formulas on points of subdivision.

DOI: https://doi.org/10.22405/2226-8383-2018-21-3-18-28

Full text: PDF file (688 kB)

UDC: 511.3
Accepted:22.10.2020

Citation: A. A. Artemov, V. N. Chubarikov, “On the trigonometric sum modulo subdivision of the real axis”, Chebyshevskii Sb., 21:3 (2020), 18–28

Citation in format AMSBIB
\Bibitem{ArtChu20} \by A.~A.~Artemov, V.~N.~Chubarikov \paper On the trigonometric sum modulo subdivision of the real axis \jour Chebyshevskii Sb. \yr 2020 \vol 21 \issue 3 \pages 18--28 \mathnet{http://mi.mathnet.ru/cheb924} \crossref{https://doi.org/10.22405/2226-8383-2018-21-3-18-28}