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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 Weil–Brezin Map ： ウィキペディア英語版 Weil–Brezin Map In mathematics, the Weil–Brezin map, named after André Weil〔Weil, André. "Sur certains groupes d'opérateurs unitaires." Acta mathematica 111.1 (1964): 143-211.〕 and Jonathan Brezin,〔Brezin, Jonathan. "Harmonic analysis on nilmanifolds." Transactions of the American Mathematical Society 150.2 (1970): 611-618.〕 is a unitary transformation that maps a Schwartz function on the real line to a smooth function on the Heisenberg manifold. The Weil–Brezin map gives a geometric interpretation of the Fourier transform, the Plancherel theorem and the Poisson summation formula.〔Auslander, Louis, and Richard Tolimieri. Abelian harmonic analysis, theta functions and function algebras on a nilmanifold. Springer, 1975.〕〔Auslander, Louis. "Lecture notes on nil-theta functions." Conference Board of the Mathematical Sciences, 1977.〕〔Zhang, D. ("Integer Linear Canonical Transforms, Their Discretization, and Poisson Summation Formulae" )〕 The image of Gaussian functions under the Weil–Brezin map are nil-theta functions, which are related to theta functions. The Weil–Brezin map is sometimes referred to as the Zak transform,〔http://mathworld.wolfram.com/ZakTransform.html〕 which is widely applied in the field of physics and signal processing; however, the Weil–Brezin Map is defined via Heisenberg group geometrically, whereas there is no direct geometric or group theoretic interpretation from the Zak transform. ==Heisenberg manifold== The (continuous) Heisenberg group $N$ is the 3-dimensional Lie group that can be represented by triples of real numbers with multiplication rule : $\backslash langle\; x,y,t\backslash rangle\; \backslash langle\; a,b,c\backslash rangle\; =\; \backslash langle\; x+a,\; y+b,\; t+c+xb\backslash rangle.$ The discrete Heisenberg group $\backslash Gamma$ is the discrete subgroup of $N$ whose elements are represented by the triples of integers. Considering $\backslash Gamma$ acts on $N$ on the left, the quotient manifold $\backslash Gamma\backslash backslash\; N$ is called the Heisenberg manifold. The Heisenberg group acts on the Heisenberg manifold on the right. The Haar measure $\backslash mu\; =\; dx\; \backslash wedge\; dy\; \backslash wedge\; dt$ on the Heisenberg group induces a right-translation-invariant measure on the Heisenberg manifold. The space of complex-valued square integrable functions on the Heisenberg manifold has a right-translation-invariant orthogonal decomposition: :$L^2(\backslash Gamma\backslash backslash\; N)\; =\; \backslash oplus\_$. 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「Weil–Brezin Map」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.