ATS theorem

Certain notations

[1]. For ${\displaystyle B>0,B\to +\infty ,}$ or ${\displaystyle B\to 0,}$ the record

${\displaystyle 1\ll {\frac {A}{B}}\ll 1}$

means that there are the constants ${\displaystyle C_{1}>0}$

and ${\displaystyle C_{2}>0,}$

such that

${\displaystyle C_{1}\leq {\frac {|A|}{B}}\leq C_{2}.}$

[2]. For a real number ${\displaystyle \alpha ,}$ the record ${\displaystyle \|\alpha \|}$ means that

${\displaystyle \|\alpha \|=\min(\{\alpha \},1-\{\alpha \}),}$

where

${\displaystyle \{\alpha \}}$

is the fractional part of ${\displaystyle \alpha .}$

ATS theorem

Let the real functions ƒ(x) and ${\displaystyle \varphi (x)}$ satisfy on the segment [a, b] the following conditions:

1) ${\displaystyle f''''(x)}$ and ${\displaystyle \varphi ''(x)}$ are continuous;

2) there exist numbers ${\displaystyle H,}$ ${\displaystyle U}$ and ${\displaystyle V}$ such that

${\displaystyle H>0,\qquad 1\ll U\ll V,\qquad 0<b-a\leq V}$

and

${\displaystyle {\begin{array}{rc}{\frac {1}{U}}\ll f''(x)\ll {\frac {1}{U}}\ ,&\varphi (x)\ll H,\\\\f'''(x)\ll {\frac {1}{UV}}\ ,&\varphi '(x)\ll {\frac {H}{V}},\\\\f''''(x)\ll {\frac {1}{UV^{2}}}\ ,&\varphi ''(x)\ll {\frac {H}{V^{2}}}.\\\\\end{array}}}$

Then, if we define the numbers ${\displaystyle x_{\mu }}$ from the equation

${\displaystyle f'(x_{\mu })=\mu ,}$

we have

${\displaystyle \sum _{a<\mu \leq b}\varphi (\mu )e^{2\pi if(\mu )}=\sum _{f'(a)\leq \mu \leq f'(b)}C(\mu )Z(\mu )+R,}$

where

${\displaystyle R=O\left({\frac {HU}{b-a}}+HT_{a}+HT_{b}+H\log \left(f'(b)-f'(a)+2\right)\right);}$

${\displaystyle T_{j}={\begin{cases}0,&{\text{if }}f'(j){\text{ is an integer}};\\\min \left({\frac {1}{\|f'(j)\|}},{\sqrt {U}}\right),&{\text{if }}\|f'(j)\|\neq 0;\\\end{cases}}}$

${\displaystyle j=a,b;}$

${\displaystyle C(\mu )={\begin{cases}1,&{\text{if }}f'(a)<\mu <f'(b);\\{\frac {1}{2}},&{\text{if }}\mu =f'(a){\text{ or }}\mu =f'(b);\\\end{cases}}}$

${\displaystyle Z(\mu )={\frac {1+i}{\sqrt {2}}}{\frac {\varphi (x_{\mu })}{\sqrt {f''(x_{\mu })}}}e^{2\pi i(f(x_{\mu })-\mu x_{\mu })}\ .}$

The most simple variant of the formulated theorem is the statement, which is called in the literature the Van der Corput lemma.

Van der Corput lemma

Let ${\displaystyle f}$ be a real differentiable function in the interval ${\displaystyle ]a,b],}$ moreover, inside of this interval, its derivative ${\displaystyle f'}$ is a monotonic and a sign-preserving function, and for the constant ${\displaystyle \delta }$ such that ${\displaystyle 0<\delta <1}$ satisfies the inequality ${\displaystyle |f'|\leq \delta .}$ Then

${\displaystyle \sum _{a<k\leq b}e^{2\pi if(k)}=\int _{a}^{b}e^{2\pi if(x)}dx+\theta \left(3+{\frac {2\delta }{1-\delta }}\right),}$

where ${\displaystyle |\theta |\leq 1.}$

Remark

If the parameters ${\displaystyle a}$ and ${\displaystyle b}$ are integers, then it is possible to substitute the last relation by the following ones:

${\displaystyle \sum _{a<k\leq b}e^{2\pi if(k)}=\int _{a}^{b}e^{2\pi if(x)}\,dx+{\frac {1}{2}}e^{2\pi if(b)}-{\frac {1}{2}}e^{2\pi if(a)}+\theta {\frac {2\delta }{1-\delta }},}$

where ${\displaystyle |\theta |\leq 1.}$

Additional sources

On the applications of ATS to the problems of physics see: