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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 Coorbit theory ： ウィキペディア英語版 Coorbit theory In mathematics, coorbit theory was developed by Hans Georg Feichtinger and Karlheinz Gröchenig around 1990.〔H. G. Feichtinger and K. Gröchenig. "A unified approach to atomic decompositions via integrable group representations" Lect. Notes in Math. 1302:52—73, 1988.〕〔H. G. Feichtinger and K. Gröchenig. "Banach spaces related to integrable group representations and their atomic decompositions, I" J. Funct. Anal. 86(2):307–340, 1989.〕〔H. G. Feichtinger and K. Gröchenig. "Banach spaces related to integrable group representations and their atomic decompositions, II" Monatsh. Math. 108(2-3):129–148, 1989.〕 It provides theory for atomic decomposition of a range of Banach spaces of distributions. Among others the well established wavelet transform and the short-time Fourier transform are covered by the theory. The starting point is a square integrable representation $\backslash pi$ of a locally compact group $\backslash mathcal\; G$ on a Hilbert space $\backslash mathcal\; H$, with which one can define a transform of a function $f\; \backslash in\; \backslash mathcal\; H$ with respect to $g\backslash in\; \backslash mathcal\; H$ by $V\_g\; f\; (x)\; =\; \backslash langle\; f,\; \backslash pi(x)g\; \backslash rangle$. Many important transforms are special cases of the transform, e.g. the short-time Fourier transform and the wavelet transform for the Heisenberg group and the affine group respectively. Representation theory yields the reproducing formula $V\_g\; f\; =\; V\_g\; f$ * V_g g. By discretization of this continuous convolution integral it can be shown that by sufficiently dense sampling in phase space the corresponding functions will span a frame for the Hilbert space. An important aspect of the theory is the derivation of atomic decompositions for Banach spaces. One of the key steps is to define the voice transform for distributions in a natural way. For a given Banach space $Y$, the corresponding coorbit space is defined as the set of all distributions such that $V\_g\; f\; \backslash in\; Y$. The reproducing formula is true also in this case and therefore it is possible to obtain atomic decompositions for coorbit spaces. ==References== 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「Coorbit theory」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.