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
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developmen ...
, 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
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...
in the analysis of the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
, in the solution of problems connected with
integer points in the domains on plane and in space, in the study of the
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
, 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
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in m ...
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
Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
and
J. E. Littlewood
John Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician. He worked on topics relating to analysis, number theory, and differential equations, and had lengthy collaborations with G. H. Hardy, Srinivasa Ramanu ...
,
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,).
Certain notations
''For''
''or''
''the record''
::
: '' means that there are the constants''
: ''and''
: ''such that''
::
''For a real number''
''the record''
''means that''
::
:''where''
::
:''is the fractional part of''
ATS theorem
''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''
::
:''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
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,.
Notes
{{DEFAULTSORT:Ats Theorem
Theorems in analysis