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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 \scriptstyle \psi \,\in\, L^2(\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 \scriptstyle L^2\left(\mathbb\right) of square integrable functions.
The Hilbert basis is constructed as the family of functions \scriptstyle \ by means of dyadic translations and dilations of \scriptstyle \psi\,,
:\psi_(x) = 2^\frac \psi\left(2^jx - k\right)\,
for integers \scriptstyle j,\, k \,\in\, \mathbb.
If under the standard inner product on \scriptstyle L^2\left(\mathbb\right),
:\langle f, g\rangle = \int_^\infty f(x)\overlinedx
this family is orthonormal, it is an orthonormal system:
:\begin
\langle\psi_,\psi_\rangle &= \int_^\infty \psi_(x)\overlinedx \\
&=\delta_\delta_
\end
where \scriptstyle \delta_\, is the Kronecker delta.
Completeness is satisfied if every function \scriptstyle h \,\in\, L^2\left(\mathbb\right) may be expanded in the basis as
:h(x) = \sum_^\infty c_ \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



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