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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 Wavelet transform ： ウィキペディア英語版 Wavelet transform In mathematics, a wavelet series is a representation of a square-integrable (real- or complex-valued) function by a certain orthonormal series generated by a wavelet. Nowadays, wavelet transformation is one of the most popular of the time-frequency-transformations. This article provides a formal, mathematical definition of an orthonormal wavelet and of the integral wavelet transform. ==Definition== A function $\backslash scriptstyle\; \backslash psi\; \backslash ,\backslash in\backslash ,\; L^2(\backslash mathbb)$ is called an orthonormal wavelet if it can be used to define a Hilbert basis, that is a complete orthonormal system, for the Hilbert space $\backslash scriptstyle\; L^2\backslash left(\backslash mathbb\backslash right)$ of square integrable functions. The Hilbert basis is constructed as the family of functions $\backslash scriptstyle\; \backslash$ by means of dyadic translations and dilations of $\backslash scriptstyle\; \backslash psi\backslash ,$, :$\backslash psi\_(x)\; =\; 2^\backslash frac\; \backslash psi\backslash left(2^jx\; -\; k\backslash right)\backslash ,$ for integers $\backslash scriptstyle\; j,\backslash ,\; k\; \backslash ,\backslash in\backslash ,\; \backslash mathbb$. If under the standard inner product on $\backslash scriptstyle\; L^2\backslash left(\backslash mathbb\backslash right)$, :$\backslash langle\; f,\; g\backslash rangle\; =\; \backslash int\_^\backslash infty\; f(x)\backslash overlinedx$ this family is orthonormal, it is an orthonormal system: :$\backslash begin$ \langle\psi_,\psi_\rangle &= \int_^\infty \psi_(x)\overlinedx \\ &=\delta_\delta_ \end where $\backslash scriptstyle\; \backslash delta\_\backslash ,$ is the Kronecker delta. Completeness is satisfied if every function $\backslash scriptstyle\; h\; \backslash ,\backslash in\backslash ,\; L^2\backslash left(\backslash mathbb\backslash right)$ may be expanded in the basis as :$h(x)\; =\; \backslash sum\_^\backslash infty\; c\_\; \backslash psi\_(x)$ with convergence of the series understood to be convergence in norm. Such a representation of a function ''f'' is known as a wavelet series. This implies that an orthonormal wavelet is self-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「Wavelet transform」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.