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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 Van der Corput lemma (harmonic analysis) ： ウィキペディア英語版 Van der Corput lemma (harmonic analysis) In mathematics, in the field of harmonic analysis, the van der Corput lemma is an estimate for oscillatory integrals named after the Dutch mathematician J. G. van der Corput. The following result is stated by E. Stein:〔Elias Stein, ''Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals''. Princeton University Press, 1993. ISBN 0-691-03216-5〕 Suppose that a real-valued function $\backslash phi(x)\backslash ,$ is smooth in an open interval $(a,b)\backslash ,$, and that $|\backslash phi^(x)|\backslash ge\; 1$ for all $x\backslash in\; (a,b)$. Assume that either $k\backslash ge\; 2$, or that $k=1\backslash ,$ and $\backslash phi\text{'}(x)\backslash ,$ is monotone for $x\backslash in\backslash R$. There is a constant $c\_k\backslash ,$, which does not depend on $\backslash phi\backslash ,$, such that :$$ \Big|\int_a^b e^\Big|\le c_k\lambda^, for any $\backslash lambda\backslash in\backslash R$. ==Sublevel set estimates== The van der Corput lemma is closely related to the sublevel set estimates (see for example 〔M. Christ, ''Hilbert transforms along curves'', Ann. of Math. 122 (1985), 575--596〕), which give the upper bound on the measure of the set where a function takes values not larger than $\backslash epsilon\backslash ,$. Suppose that a real-valued function $\backslash phi(x)\backslash ,$ is smooth on a finite or infinite interval $I\backslash subset\backslash R$, and that $|\backslash phi^(x)|\backslash ge\; 1\backslash ,$ for all $x\backslash in\; I$. There is a constant $c\_k\backslash ,$, which does not depend on $\backslash phi\backslash ,$, such that for any $\backslash epsilon\backslash ge\; 0\backslash ,$ the measure of the sublevel set $\backslash$ is bounded by $c\_k\backslash epsilon^\backslash ,$. 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「Van der Corput lemma (harmonic analysis)」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.