ATS theorem

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

: $$

S = \sum_{a

are under study.

Here $\backslash varphi(x)$ and $f(x)$ are real valued functions of a real

argument, and $i^2=\; -1.$

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 $S$

to be the number $b-a$

(for the integers $a$ and $b,$ this is the number of the summands in $S$).

Under certain conditions on $\backslash varphi(x)$ and $f(x)$

the sum $S$ can be

substituted with good accuracy by another sum $S\_1,$

: $$

S_1 = \sum_{\alpha

where the length $\backslash beta-\backslash alpha$ is far less than $b-a.$

First relations of the form

: $$

S = S_1 + R , \qquad (3)

where $S\; ,$ $S\_1$ are the sums (1) and (2) respectively, $R$ is

a remainder term, with concrete functions $\backslash varphi(x)$ and $f(x),$

were obtained by G. H. Hardy and J. E. Littlewood,{{cite journal | last1=Hardy | first1=G. H. | last2=Littlewood | first2=J. E. | title=Some problems of diophantine approximation: Part II. The trigonometrical series associated with the elliptic θ-functions | journal=Acta Mathematica | publisher=International Press of Boston | volume=37 | year=1914 | issn=0001-5962 | doi=10.1007/bf02401834 | pages=193–239|doi-access=free}}{{cite journal | last1=Hardy | first1=G. H. | last2=Littlewood | first2=J. E. | title=Contributions to the theory of the riemann zeta-function and the theory of the distribution of primes | journal=Acta Mathematica | publisher=International Press of Boston | volume=41 | year=1916 | issn=0001-5962 | doi=10.1007/bf02422942 | pages=119–196|doi-access=free}}{{cite journal | last1=Hardy | first1=G. H. | last2=Littlewood | first2=J. E. | title=The zeros of Riemann's zeta-function on the critical line | journal=Mathematische Zeitschrift | publisher=Springer Science and Business Media LLC | volume=10 | issue=3–4 | year=1921 | issn=0025-5874 | doi=10.1007/bf01211614 | pages=283–317| s2cid=126338046 | url=https://zenodo.org/record/1447415 }}

when they

deduced approximate functional equation for the Riemann zeta function

$\backslash zeta(s)$ and by I. M. Vinogradov,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). 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,{{cite journal | last=van der Corput | first=J. G. | title=Zahlentheoretische Abschätzungen | journal=Mathematische Annalen | publisher=Springer Science and Business Media LLC | volume=84 | issue=1–2 | year=1921 | issn=0025-5831 | doi=10.1007/bf01458693 | pages=53–79 | s2cid=179178113 | language=de}}{{cite journal | last=van der Corput | first=J. G. | title=Verschärfung der Abschätzung beim Teilerproblem | journal=Mathematische Annalen | publisher=Springer Science and Business Media LLC | volume=87 | issue=1–2 | year=1922 | issn=0025-5831 | doi=10.1007/bf01458035 | pages=39–65 | s2cid=177789678 | language=de}} (on the recent

results connected with the Van der Corput theorem one can read at

{{cite book | last=Montgomery | first=Hugh | title=Ten lectures on the interface between analytic number theory and harmonic analysis | publisher=Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society | location=Providence, R.I | year=1994 | isbn=978-0-8218-0737-8 | oclc=30811108 }}).

In every one of the above-mentioned works,

some restrictions on the functions

$\backslash varphi(x)$ and $f(x)$ were imposed. With

convenient (for applications) restrictions on

$\backslash varphi(x)$ and $f(x),$ the theorem was proved by A. A. Karatsuba in {{cite journal | last=Karatsuba | first=A. A. | title=Approximation of exponential sums by shorter ones | journal=Proceedings of the Indian Academy of Sciences, Section A | publisher=Springer Science and Business Media LLC | volume=97 | issue=1–3 | year=1987 | issn=0370-0089 | doi=10.1007/bf02837821 | pages=167–178| s2cid=120389154 }} (see also,A. A. Karatsuba, S. M. Voronin. The Riemann Zeta-Function. (W. de Gruyter, Verlag: Berlin, 1992).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).).

[1]. For $B\; >\; 0,\; B\; \backslash to\; +\backslash infty,$

or $B\; \backslash to\; 0,$ the record

:: $1\; \backslash ll\; \backslash frac\{A\}\{B\}\; \backslash ll\; 1$

: means that there are the constants $C\_1\; >\; 0$

: and $C\_2\; >\; 0,$

: such that

:: $C\_1\; \backslash leq\backslash frac$

A

{B} \leq C_2.

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

:: $\backslash |\backslash alpha\backslash |\; =\; \backslash min(\backslash \{\backslash alpha\backslash \},1-\; \backslash \{\backslash alpha\backslash \}),$

:where

:: $\backslash \{\backslash alpha\backslash \}$

:is the fractional part of $\backslash alpha.$

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

1) $f$(x) and $\backslash varphi$(x) are continuous;''

2) there exist numbers

$H,$ $U$ and $V$ such that

::

:and

:: $$

\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 $x\_\backslash mu$ from the equation

: $$

f'(x_\mu) = \mu,

we have

: $$

\sum_{a< \mu\le b} \varphi(\mu)e^{2\pi i f(\mu)} = \sum_{f'(a)\le\mu\le

f'(b)}C(\mu)Z(\mu) + R ,

where

: $$

R = O

\left(\frac{HU}{b-a} + HT_a + HT_b +

H\log\left(f'(b)-f'(a)+2\right)\right);

: $$

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)\| \ne 0; \\

\end{cases}

$j\; =\; a,b;$

: $$

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}

: $$

Z(\mu) = \frac{1+i}{\sqrt2} \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.

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

: $$

\sum_{a

\theta\left(3 + \frac{2\delta}{1-\delta}\right),

where $|\backslash theta|\; \backslash le\; 1.$

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

: $$

\sum_{a

where $|\backslash theta|\; \backslash le\; 1.$

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

• {{cite journal | last=Karatsuba | first=Ekatherina A. | title=Approximation of sums of oscillating summands in certain physical problems | journal=Journal of Mathematical Physics | publisher=AIP Publishing | volume=45 | issue=11 | year=2004 | issn=0022-2488 | doi=10.1063/1.1797552 | pages=4310–4321}}
• {{cite journal | last=Karatsuba | first=Ekatherina A. | title=On an approach to the study of the Jaynes–Cummings sum in quantum optics | journal=Numerical Algorithms | publisher=Springer Science and Business Media LLC | volume=45 | issue=1–4 | date=2007-07-20 | issn=1017-1398 | doi=10.1007/s11075-007-9070-x | pages=127–137| s2cid=13485016 }}
• {{cite journal | last1=Chassande-Mottin | first1=Éric | last2=Pai | first2=Archana | title=Best chirplet chain: Near-optimal detection of gravitational wave chirps | journal=Physical Review D | publisher=American Physical Society (APS) | volume=73 | issue=4 | date=2006-02-27 | issn=1550-7998 | doi=10.1103/physrevd.73.042003 | arxiv=gr-qc/0512137 | page=042003| hdl=11858/00-001M-0000-0013-4BBD-B | s2cid=56344234 | hdl-access=free }}
• {{cite journal | last1=Fleischhauer | first1=M. | last2=Schleich | first2=W. P. | title=Revivals made simple: Poisson summation formula as a key to the revivals in the Jaynes-Cummings model | journal=Physical Review A | publisher=American Physical Society (APS) | volume=47 | issue=5 | date=1993-05-01 | issn=1050-2947 | doi=10.1103/physreva.47.4258 | pages=4258–4269| pmid=9909432 }}

{{DEFAULTSORT:Ats Theorem}}

Category:Theorems in mathematical analysis