On the trigonometric sum modulo subdivision of the real axis

A. A. Artemov ; V. N. Chubarikov

Čebyševskij sbornik, Tome 21 (2020) no. 3, pp. 18-28.

Résumé

The estimate of the trigonometric sum of the kind $$ S=\sum_{a\leq b}e^{2\pi if(t_s)}, $$ 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 $$ 0=t_0\dots\dots, \lim\limits_{n\to\infty}t_n=+\infty, $$ 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 $t_s\leq x$ The fractional part $\{x\}_{\Delta}$ is defined by the equality $$ \{x\}_{\Delta}=\frac{x-t_s}{t_{s+1}-t_s}, $$ if $t_s\leq x$ 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 $\Delta=\{t_s\}, 0=t_0$ 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 $02\delta\delta^{-1}(a,b)1$ and that for all $x$ from this interval is valid inequality $|f'(x)|\leq\delta.$ Then we have $$ \sum_{a\leq b}e^{2\pi if(t_s)}=\int\limits_{a}^{b}\rho'_\Delta(x)e^{2\pi if(x)} dx+10\theta\frac{\delta}{1-\delta\delta^{-1}(a,b)}, |\theta|\leq 1. $$

Détail

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.

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