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 (1) by a suitable function was studied already by Euler 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 and J. E. Littlewood, when they deduced approximate functional equation for the Riemann zeta function $\zeta(s)$ 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,).
Read more about this topic: ATS Theorem
Famous quotes containing the words history of, history and/or problem:
“Culture, the acquainting ourselves with the best that has been known and said in the world, and thus with the history of the human spirit.”
—Matthew Arnold (18221888)
“Free from public debt, at peace with all the world, and with no complicated interests to consult in our intercourse with foreign powers, the present may be hailed as the epoch in our history the most favorable for the settlement of those principles in our domestic policy which shall be best calculated to give stability to our Republic and secure the blessings of freedom to our citizens.”
—Andrew Jackson (17671845)
“The family environment in which your children are growing up is different from that in which you grew up. The decisions our parents made and the strategies they used were developed in a different context from what we face today, even if the content of the problem is the same. It is a mistake to think that our own experience as children and adolescents will give us all we need to help our children. The rules of the game have changed.”
—Lawrence Kutner (20th century)