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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 Modulation space ： ウィキペディア英語版 Modulation space Modulation spaces〔Foundations of Time-Frequency Analysis by Karlheinz Gröchenig〕 are a family of Banach spaces defined by the behavior of the short-time Fourier transform with respect to a test function from the Schwartz space. They were originally proposed by Hans Georg Feichtinger and are recognized to be ''the right kind of function spaces'' for time-frequency analysis. ''Feichtinger's algebra'', while originally introduced as a new Segal algebra,〔H. Feichtinger. "On a new Segal algebra" Monatsh. Math. 92:269–289, 1981.〕 is identical to a certain modulation space and has become a widely used space of test functions for time-frequency analysis. Modulation spaces are defined as follows. For $1\backslash leq\; p,q\; \backslash leq\; \backslash infty$, a non-negative function $m(x,\backslash omega)$ on $\backslash mathbb^$ and a test function $g\; \backslash in\; \backslash mathcal(\backslash mathbb^d)$, the modulation space $M^\_m(\backslash mathbb^d)$ is defined by : In the above equation, $V\_gf$ denotes the short-time Fourier transform of $f$ with respect to $g$ evaluated at $(x,\backslash omega)$, namely :$V\_gf(x,\backslash omega)=\backslash int\_f(t)\backslash overlinee^dt=\backslash mathcal^\_(\backslash overline\backslash hat(\backslash xi+\backslash omega))(x).$ In other words, $f\backslash in\; M^\_m(\backslash mathbb^d)$ is equivalent to $V\_gf\backslash in\; L^\_m(\backslash mathbb^)$. The space $M^\_m(\backslash mathbb^d)$ is the same, independent of the test function $g\; \backslash in\; \backslash mathcal(\backslash mathbb^d)$ chosen. The canonical choice is a Gaussian. We also have a Besov-type definition of modulation spaces as follows.〔B.X. Wang, Z.H. Huo, C.C. Hao, and Z.H. Guo. Harmonic Analysis Method for Nonlinear Evolution Equations. World Scientific, 2011.〕 :, where $\backslash$ is a suitable unity partition. If $m(x,\backslash omega)=\backslash langle\; \backslash omega\backslash rangle^s$, then $M^s\_=M^\_m$. == Feichtinger's algebra == For $p=q=1$ and $m(x,\backslash omega)\; =\; 1$, the modulation space $M^\_m(\backslash mathbb^d)\; =\; M^1(\backslash mathbb^d)$ is known by the name Feichtinger's algebra and often denoted by $S\_0$ for being the minimal Segal algebra invariant under time-frequency shifts, i.e. combined translation and modulation operators. $M^1(\backslash mathbb^d)$ is a Banach space embedded in $L^1(\backslash mathbb^d)\; \backslash cap\; C\_0(\backslash mathbb^d)$, and is invariant under the Fourier transform. It is for these and more properties that $M^1(\backslash mathbb^d)$ is a natural choice of test function space for time-frequency analysis. Fourier transform $\backslash mathcal$ is an automorphism on $M^$. 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「Modulation space」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.