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
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.
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, 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, (on the recent results connected with the Van der Corput theorem one can read at ).
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  (see also,).
. For or the record
means that there are the constants and such that
. For a real number the record means that
is the fractional part of
Let the real functions ƒ(x) and satisfy on the segment [a, b] the following conditions:
1) and are continuous;
2) there exist numbers and such that
Then, if we define the numbers from the equation
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
If the parameters and are integers, then it is possible to substitute the last relation by the following ones:
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