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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 dual wavelet ： ウィキペディア英語版 dual wavelet In mathematics, a dual wavelet is the dual to a wavelet. In general, the wavelet series generated by a square integrable function will have a dual series, in the sense of the Riesz representation theorem. However, the dual series is not in general representable by a square integral function itself. ==Definition== Given a square integrable function $\backslash psi\backslash in\; L^2(\backslash mathbb)$, define the series $\backslash$ by :$\backslash psi\_(x)\; =\; 2^\backslash psi(2^jx-k)$ for integers $j,k\backslash in\; \backslash mathbb$. Such a function is called an ''R''-function if the linear span of $\backslash$ is dense in $L^2(\backslash mathbb)$, and if there exist positive constants ''A'', ''B'' with :$A\; \backslash Vert\; c\_\; \backslash Vert^2\_\; \backslash leq$ \bigg\Vert \sum_^\infty c_\psi_\bigg\Vert^2_ \leq B \Vert c_ \Vert^2_\, for all bi-infinite square summable series $\backslash$. Here, $\backslash Vert\; \backslash cdot\; \backslash Vert\_$ denotes the square-sum norm: :$\backslash Vert\; c\_\; \backslash Vert^2\_\; =\; \backslash sum\_^\backslash infty\; \backslash vert\; c\_\backslash vert^2$ and $\backslash Vert\; \backslash cdot\backslash Vert\_$ denotes the usual norm on $L^2(\backslash mathbb)$: :$\backslash Vert\; f\backslash Vert^2\_=\; \backslash int\_^\backslash infty\; \backslash vert\; f(x)\backslash vert^2\; dx$ By the Riesz representation theorem, there exists a unique dual basis $\backslash psi^$ such that :$\backslash langle\; \backslash psi^\; \backslash vert\; \backslash psi\_\; \backslash rangle\; =\; \backslash delta\_\; \backslash delta\_$ where $\backslash delta\_$ is the Kronecker delta and $\backslash langle\; f\backslash vert\; g\; \backslash rangle$ is the usual inner product on $L^2(\backslash mathbb)$. Indeed, there exists a unique series representation for a square integrable function ''f'' expressed in this basis: :$f(x)\; =\; \backslash sum\_\; \backslash langle\; \backslash psi^\; \backslash vert\; f\; \backslash rangle\; \backslash psi\_(x)$ If there exists a function $\backslash tilde\; \backslash in\; L^2(\backslash mathbb)$ such that :$\backslash tilde\_\; =\; \backslash psi^$ then $\backslash tilde$ is called the dual wavelet or the wavelet dual to ψ. In general, for some given ''R''-function ψ, the dual will not exist. In the special case of $\backslash psi\; =\; \backslash tilde$, the wavelet is said to be an orthogonal wavelet. An example of an ''R''-function without a dual is easy to construct. Let $\backslash phi$ be an orthogonal wavelet. Then define $\backslash psi(x)\; =\; \backslash phi(x)\; +\; z\backslash phi(2x)$ for some complex number ''z''. It is straightforward to show that this ψ does not have a wavelet dual. 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「dual wavelet」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.