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

are under study.

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 ).

First relations of the form

were obtained by G. H. Hardy and J. E. Littlewood, and by I. M. Vinogradov, 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, (on the recent results connected with the Van der Corput theorem one can read at ).

the theorem was proved by A. A. Karatsuba in (see also,).

Certain notations

[1]. For or the record

1. means that there are the constants

2. and

3. such that

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

1. where

2. is the fractional part of

ATS theorem

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

and are continuous;

2) there exist numbers and such that

1. 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 lemma

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

Remark

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

Additional sources

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

Notes