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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク ATS theorem ： ウィキペディア英語版 ATS theorem 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, sums of the form : $$ S = \sum_ \varphi(k)e^ \ \ \ (1) are under study. Here $\backslash varphi(x)$ and $f(x)$ are real valued functions of a real argument, and $i^2=\; -1.$ Such sums appear, for example, in number theory in the analysis of the Riemann zeta function, in the solution of problems connected with integer points in the domains on plane and in space, in the study of the Fourier series, and in the solution of such differential equations as the wave equation, the potential equation, the heat conductivity equation. The problem of approximation of the series (1) by a suitable function was studied already by Euler and Poisson. We shall define the length of the sum $S$ to be the number $b-a$ (for the integers $a$ and $b,$ this is the number of the summands in $S$). Under certain conditions on $\backslash varphi(x)$ and $f(x)$ the sum $S$ can be substituted with good accuracy by another sum $S\_1,$ : $$ S_1 = \sum_ \Phi(k)e^ , \ \ \ (2) where the length $\backslash beta-\backslash alpha$ is far less than $b-a.$ First relations of the form : $$ S = S_1 + R , \ \ \ (3) where $S\; ,$ $S\_1$ are the sums (1) and (2) respectively, $R$ is a remainder term, with concrete functions $\backslash varphi(x)$ and $f(x),$ were obtained by G. H. Hardy and J. E. Littlewood,〔G.~H. Hardy and J.~E. Littlewood. The trigonometrical series associated with the elliptic $\theta$-functions. Acta Math. 37, pp. 193—239 (1914).〕〔G.~H. Hardy and J.~E. Littlewood. Contributions to the theory of Riemann Zeta-Function and the theory of the distribution of primes. Acta Math. 41, pp. 119—196 (1918).〕〔G.~H. Hardy and J.~E. Littlewood. The zeros of Riemann's zeta-function on the critical line, Math. Z., 10, pp. 283–317 (1921).〕 when they deduced approximate functional equation for the Riemann zeta function $\backslash zeta(s)$ and by I. M. Vinogradov,〔I.~M. Vinogradov. On the average value of the number of classes of purely root form of the negative determinant Communic. of Khar. Math. Soc., ''16'', 10–38 (1917).〕 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,〔J.~G. Van der Corput. Zahlentheoretische Abschätzungen. Math. Ann. 84, pp. 53–79 (1921).〕〔J.~G. Van der Corput. Verschärfung der Abschätzung beim Teilerproblem. Math. Ann., 87, pp. 39–65 (1922).〕 (on the recent results connected with the Van der Corput theorem one can read at 〔H.~L. Montgomery. Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Am. Math. Soc., 1994.〕). In every one of the above-mentioned works, some restrictions on the functions $\backslash varphi(x)$ and $f(x)$ were imposed. With convenient (for applications) restrictions on $\backslash varphi(x)$ and $f(x),$ the theorem was proved by A. A. Karatsuba in 〔A.~A. Karatsuba. Approximation of exponential sums by shorter ones. Proc. Indian. Acad. Sci. (Math. Sci.) 97: 1–3, pp. 167—178 (1987). 〕 (see also,〔A.~A. Karatsuba, S. M. Voronin. The Riemann Zeta-Function. (W. de Gruyter, Verlag: Berlin, 1992).〕〔A.~A. Karatsuba, M. A. Korolev. The theorem on the approximation of a trigonometric sum by a shorter one. Izv. Ross. Akad. Nauk, Ser. Mat. 71:3, pp. 63—84 (2007).〕). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「ATS theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.