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.

Contents

History of the problemEdit

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

 

are under study.

Here   and   are real valued functions of a real argument, and   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   to be the number   (for the integers   and   this is the number of the summands in  ).

Under certain conditions on   and   the sum   can be substituted with good accuracy by another sum  

 

where the length   is far less than  

First relations of the form

 

where     are the sums (1) and (2) respectively,   is a remainder term, with concrete functions   and   were obtained by G. H. Hardy and J. E. Littlewood,[1][2][3] when they deduced approximate functional equation for the Riemann zeta function   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   and   were imposed. With convenient (for applications) restrictions on   and   the theorem was proved by A. A. Karatsuba in [8] (see also,[9][10]).

Certain notationsEdit

[1]. For   or   the record

  means that there are the constants   and   such that

 

[2]. For a real number   the record   means that

 

where

 

is the fractional part of  

ATS theoremEdit

Let the real functions ƒ(x) and   satisfy on the segment [ab] the following conditions:

1)   and   are continuous;

2) there exist numbers     and   such that

 
and
 

Then, if we define the numbers   from the equation

 

we have

 

where

 
 

 

 
 

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 lemmaEdit

Let   be a real differentiable function in the interval   moreover, inside of this interval, its derivative   is a monotonic and a sign-preserving function, and for the constant   such that   satisfies the inequality   Then

 

where  

RemarkEdit

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

 

where  

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

NotesEdit

  1. ^ G.~H. Hardy and J.~E. Littlewood. The trigonometrical series associated with the elliptic $\theta$-functions. Acta Math. 37, pp. 193—239 (1914).
  2. ^ G.~H. Hardy and J.~E. Littlewood. Contributions to the theory of Riemann Zeta-Function and the theory of the distribution of primes. Acta Math. 41, pp. 119—196 (1918).
  3. ^ G.~H. Hardy and J.~E. Littlewood. The zeros of Riemann's zeta-function on the critical line, Math. Z., 10, pp. 283–317 (1921).
  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. ^ J.~G. Van der Corput. Zahlentheoretische Abschätzungen. Math. Ann. 84, pp. 53–79 (1921).
  6. ^ J.~G. Van der Corput. Verschärfung der Abschätzung beim Teilerproblem. Math. Ann., 87, pp. 39–65 (1922).
  7. ^ H.~L. Montgomery. Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Am. Math. Soc., 1994.
  8. ^ A.~A. Karatsuba. Approximation of exponential sums by shorter ones. Proc. Indian. Acad. Sci. (Math. Sci.) 97: 1–3, pp. 167—178 (1987).
  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. ^ E.~A. Karatsuba. Approximation of sums of oscillating summands in certain physical problems. JMP 45:11, pp. 4310—4321 (2004).
  12. ^ E.~A. Karatsuba. On an approach to the study of the Jaynes–Cummings sum in quantum optics, Numerical Algorithms, Vol. 45, No. 1–4 , pp. 127–137 (2007).
  13. ^ E. Chassande-Mottin, A. Pai. Best chirplet chain: near-optimal detection of gravitational wave chirps. Phys. Rev. D 73:4, 042003, pp. 1—23 (2006).
  14. ^ M. Fleischhauer, W.~P. Schleich. Revivals made simple: Poisson summation formula as a key to the revivals in the Jaynes-Cummings model. Phys. Rev. A 47:3, pp. 4258—4269 (1993).