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.
In some fields of mathematics and mathematical physics, sums of the form are under study.
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
this is the number of the summands in
can be substituted with good accuracy by another sum
β − α
First relations of the form where
is a remainder term, with concrete functions
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
With convenient (for applications) restrictions on
the theorem was proved by A.
For a real number
means that Let the real functions ƒ(x) and
satisfy on the segment [a, b] the following conditions: 1)
are continuous; 2) there exist numbers
The most simple variant of the formulated theorem is the statement, which is called in the literature the Van der Corput lemma.
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
satisfies the inequality
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: