ATS theorem

In mathematics, the ATS theorem is the theorem on the approximation of a trigonometric sum by a shorter one. The application of the ATS theorem in certain problems of mathematical and theoretical physics can be very helpful.

History of the problem

In some fields of mathematics and mathematical physics, sums of the form

${\displaystyle S=\sum _{a<k\leq b}\varphi (k)e^{2\pi if(k)}\qquad (1)}$

are under study.

Here ${\displaystyle \varphi (x)}$ and ${\displaystyle f(x)}$ are real valued functions of a real argument, and ${\displaystyle i^{2}=-1.}$ Such sums appear, for example, in number theory in the analysis of the Riemann zeta function, in the solution of problems connected with integer points in the domains on plane and in space, in the study of the Fourier series, and in the solution of such differential equations as the wave equation, the potential equation, the heat conductivity equation.

The problem of approximation of the series (1) by a suitable function was studied already by Euler and Poisson.

We shall define the length of the sum ${\displaystyle S}$ to be the number ${\displaystyle b-a}$ (for the integers ${\displaystyle a}$ and ${\displaystyle b,}$ this is the number of the summands in ${\displaystyle S}$).

Under certain conditions on ${\displaystyle \varphi (x)}$ and ${\displaystyle f(x)}$ the sum ${\displaystyle S}$ can be substituted with good accuracy by another sum ${\displaystyle S_{1},}$

${\displaystyle S_{1}=\sum _{\alpha <k\leq \beta }\Phi (k)e^{2\pi iF(k)},\ \ \ (2)}$

where the length ${\displaystyle \beta -\alpha }$ is far less than ${\displaystyle b-a.}$

First relations of the form

${\displaystyle S=S_{1}+R,\qquad (3)}$

where ${\displaystyle S,}$ ${\displaystyle S_{1}}$ are the sums (1) and (2) respectively, ${\displaystyle R}$ is a remainder term, with concrete functions ${\displaystyle \varphi (x)}$ and ${\displaystyle f(x),}$ were obtained by G. H. Hardy and J. E. Littlewood,[1][2][3] when they deduced approximate functional equation for the Riemann zeta function ${\displaystyle \zeta (s)}$ and by I. M. Vinogradov,[4] in the study of the amounts of integer points in the domains on plane. In general form the theorem was proved by J. Van der Corput,[5][6] (on the recent results connected with the Van der Corput theorem one can read at [7]).

In every one of the above-mentioned works, some restrictions on the functions ${\displaystyle \varphi (x)}$ and ${\displaystyle f(x)}$ were imposed. With convenient (for applications) restrictions on ${\displaystyle \varphi (x)}$ and ${\displaystyle f(x),}$ the theorem was proved by A. A. Karatsuba in [8] (see also,[9][10]).

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

On the applications of ATS to the problems of physics see,;[11][12] see also,.[13][14]

Notes

1. Hardy, G. H.; Littlewood, J. E. (1914). "Some problems of diophantine approximation: Part II. The trigonometrical series associated with the elliptic ϑ-functions". Acta Mathematica. International Press of Boston. 37 (0): 193–239. doi:10.1007/bf02401834. ISSN 0001-5962.
2. Hardy, G. H.; Littlewood, J. E. (1916). "Contributions to the theory of the riemann zeta-function and the theory of the distribution of primes". Acta Mathematica. International Press of Boston. 41 (0): 119–196. doi:10.1007/bf02422942. ISSN 0001-5962.
3. Hardy, G. H.; Littlewood, J. E. (1921). "The zeros of Riemann's zeta-function on the critical line". Mathematische Zeitschrift. Springer Science and Business Media LLC. 10 (3–4): 283–317. doi:10.1007/bf01211614. ISSN 0025-5874.
4. I. M. Vinogradov. On the average value of the number of classes of purely root form of the negative determinant Communic. of Khar. Math. Soc., 16, 10–38 (1917).
5. van der Corput, J. G. (1921). "Zahlentheoretische Abschätzungen". Mathematische Annalen (in German). Springer Science and Business Media LLC. 84 (1–2): 53–79. doi:10.1007/bf01458693. ISSN 0025-5831.
6. van der Corput, J. G. (1922). "Verschärfung der Abschätzung beim Teilerproblem". Mathematische Annalen (in German). Springer Science and Business Media LLC. 87 (1–2): 39–65. doi:10.1007/bf01458035. ISSN 0025-5831.
7. Montgomery, Hugh (1994). Ten lectures on the interface between analytic number theory and harmonic analysis. Providence, R.I: Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society. ISBN 978-0-8218-0737-8. OCLC 30811108.
8. Karatsuba, A. A. (1987). "Approximation of exponential sums by shorter ones". Proceedings of the Indian Academy of Sciences, Section A. Springer Science and Business Media LLC. 97 (1–3): 167–178. doi:10.1007/bf02837821. ISSN 0370-0089.
9. A. A. Karatsuba, S. M. Voronin. The Riemann Zeta-Function. (W. de Gruyter, Verlag: Berlin, 1992).
10. A. A. Karatsuba, M. A. Korolev. The theorem on the approximation of a trigonometric sum by a shorter one. Izv. Ross. Akad. Nauk, Ser. Mat. 71:3, pp. 63—84 (2007).
11. Karatsuba, Ekatherina A. (2004). "Approximation of sums of oscillating summands in certain physical problems". Journal of Mathematical Physics. AIP Publishing. 45 (11): 4310–4321. doi:10.1063/1.1797552. ISSN 0022-2488.
12. Karatsuba, Ekatherina A. (2007-07-20). "On an approach to the study of the Jaynes–Cummings sum in quantum optics". Numerical Algorithms. Springer Science and Business Media LLC. 45 (1–4): 127–137. doi:10.1007/s11075-007-9070-x. ISSN 1017-1398.
13. Chassande-Mottin, Éric; Pai, Archana (2006-02-27). "Best chirplet chain: Near-optimal detection of gravitational wave chirps". Physical Review D. American Physical Society (APS). 73 (4): 042003. doi:10.1103/physrevd.73.042003. hdl:11858/00-001M-0000-0013-4BBD-B. ISSN 1550-7998.
14. Fleischhauer, M.; Schleich, W. P. (1993-05-01). "Revivals made simple: Poisson summation formula as a key to the revivals in the Jaynes-Cummings model". Physical Review A. American Physical Society (APS). 47 (5): 4258–4269. doi:10.1103/physreva.47.4258. ISSN 1050-2947.