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
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 by a suitable function was studied already by Euler and
Poisson.
We shall define
the length of the sum
to be the number
.
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 and respectively, is
a remainder term, with concrete functions and
were obtained by G. H. Hardy and J. E. Littlewood,
when they
deduced approximate functional equation for the Riemann zeta function
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,.
In every one of the above-mentioned works,
some restrictions on the functions
and were imposed. With
convenient restrictions on
and the theorem was proved by A. A. Karatsuba in .

Certain notations

. For
or the record
. For a real number the record means that

ATS theorem

Let the real functions ƒ and satisfy on the segment the following conditions:
1) and are continuous;
2) there exist numbers
and such that
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

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,; see also,.