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 the, history of, history and/or problem:

    “The History of the world is not the theatre of happiness. Periods of happiness are blank pages in it, for they are periods of harmony—periods when the antithesis is in abeyance.”
    —Georg Wilhelm Friedrich Hegel (1770–1831)

    “I assure you that in our next class we will concern ourselves solely with the history of Egypt, and not with the more lurid and non-curricular subject of living mummies.”
    —Griffin Jay, and Reginald LeBorg. Prof. Norman (Frank Reicher)

    “It may be well to remember that the highest level of moral aspiration recorded in history was reached by a few ancient Jews—Micah, Isaiah, and the rest—who took no count whatever of what might not happen to them after death. It is not obvious to me why the same point should not by and by be reached by the Gentiles.”
    —Thomas Henry Huxley (1825–95)

    “War is not a life: it is a situation,
    One which may neither be ignored nor accepted,
    A problem to be met with ambush and stratagem,
    Enveloped or scattered.”
    —T.S. (Thomas Stearns)