On the trigonometric sum modulo subdivision of the real axis

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
$$
0=t_0<t_1<t_2<\dots<t_s<\dots, \lim\limits_{n\to\infty}t_n=+\infty,
$$
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 $t_s\leq x<t_{s+1}, s\geq 0.$ 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.

{\sl Let $\Delta=\{t_s\}, 0=t_0<t_1<\dots<t_s<\dots, $ 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 $\Delta_0=\{\mu_s\}, \quad \mu_s=0,5(t_s+t_{s+1}), s\geq 0,$ 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
$$\sum_{a<t_s\leq b}e^{2\pi if(t_s)}=\int\limits_{a}^{b}\rho'_\Delta(x)e^{2\pi if(x)}\,dx+10\theta\frac{\delta}{1-\delta\delta^{-1}(a,b)}, |\theta|\leq 1.
$$

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