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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 Oscillatory integral ： ウィキペディア英語版 Oscillatory integral In mathematical analysis an oscillatory integral is a type of distribution. Oscillatory integrals make rigorous many arguments that, on a naive level, appear to use divergent integrals. It is possible to represent approximate solution operators for many differential equations as oscillatory integrals. ==Definition== An oscillatory integral $f(x)$ is written formally as :$f(x)\; =\; \backslash int\; e^\backslash ,\; a(x,\backslash xi)\; \backslash ,\; \backslash mathrm\; \backslash xi$ where $\backslash phi(x,\backslash xi)$ and $a(x,\backslash xi)$ are functions defined on $\backslash mathbb\_x^n\; \backslash times\; \backslash mathrm^N\_\backslash xi$ with the following properties. :1) The function $\backslash phi$ is real valued, positive homogeneous of degree 1, and infinitely differentiable away from $\backslash$. Also, we assume that $\backslash phi$ does not have any critical points on the support of $a$. Such a function, $\backslash phi$ is usually called a phase function. In some contexts more general functions are considered, and still referred to as phase functions. :2) The function $a$ belongs to one of the symbol classes $S^m\_(\backslash mathbb\_x^n\; \backslash times\; \backslash mathrm^N\_\backslash xi)$ for some $m\; \backslash in\; \backslash mathbb$. Intuitively, these symbol classes generalize the notion of positively homogeneous functions of degree $m$. As with the phase function $\backslash phi$, in some cases the function $a$ is taken to be in more general, or just different, classes. When the formal integral defining $f(x)$ converges for all $x$ and there is no need for any further discussion of the definition of $f(x)$. However, when $m\; \backslash geq\; -\; n+1$ the oscillatory integral is still defined as a distribution on $\backslash mathbb^n$ even though the integral may not converge. In this case the distribution $f(x)$ is defined by using the fact that $a(x,\backslash xi)\; \backslash in\; S^m\_(\backslash mathbb\_x^n\; \backslash times\; \backslash mathrm^N\_\backslash xi)$ may be approximated by functions that have exponential decay in $\backslash xi$. One possible way to do this is by setting :$f(x)\; =\; \backslash lim\; \backslash limits\_\; \backslash int\; e^\backslash ,\; a(x,\backslash xi)\; e^\; \backslash ,\; \backslash mathrm\; \backslash xi$ where the limit is taken in the sense of tempered distributions. Using integration by parts it is possible to show that this limit is well defined, and that there exists a differential operator $L$ such that the resulting distribution $f(x)$ acting on any $\backslash psi$ in the Schwartz space is given by :$\backslash langle\; f,\; \backslash psi\; \backslash rangle\; =\; \backslash int\; e^\; L\; \backslash left\; (\; a(x,\backslash xi)\; \backslash ,\; \backslash psi(x)\; \backslash right\; )\; \backslash ,\; \backslash mathrm\; x\; \backslash ,\; \backslash mathrm\; \backslash xi$ where this integral converges absolutely. The operator $L$ is not uniquely defined, but can be chosen in such a way that depends only on the phase function $\backslash phi$, the order $m$ of the symbol $a$, and $N$. In fact, given any integer $M$ it is possible to find an operator $L$ so that the integrand above is bounded by $C\; (1\; +\; |\backslash xi|)^$ for $|\backslash xi|$ sufficiently large. This is the main purpose of the definition of the symbol classes. 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「Oscillatory integral」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.