ATS Theorem - History of The Problem

History of The Problem

In some fields of mathematics and mathematical physics, sums of the form

S = \sum_{a<k\le b} \varphi(k)e^{2\pi i f(k)} \ \ \ (1)

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

S_1 = \sum_{\alpha<k\le \beta} \Phi(k)e^{2\pi i F(k)}, \ \ \ (2)

where the length is far less than

First relations of the form

S = S_1 + R, \ \ \ (3)

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:

    “This is the greatest week in the history of the world since the Creation, because as a result of what happened in this week, the world is bigger, infinitely.”
    —Richard M. Nixon (1913–1995)

    “Don’t give your opinions about Art and the Purpose of Life. They are of little interest and, anyway, you can’t express them. Don’t analyse yourself. Give the relevant facts and let your readers make their own judgments. Stick to your story. It is not the most important subject in history but it is one about which you are uniquely qualified to speak.”
    —Evelyn Waugh (1903–1966)

    “What had really caused the women’s movement was the additional years of human life. At the turn of the century women’s life expectancy was forty-six; now it was nearly eighty. Our groping sense that we couldn’t live all those years in terms of motherhood alone was “the problem that had no name.” Realizing that it was not some freakish personal fault but our common problem as women had enabled us to take the first steps to change our lives.”
    —Betty Friedan (20th century)