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 itself in general representable by a square-integrable function.
Definition
Given a square-integrable function [math]\displaystyle{ \psi\in L^2(\mathbb{R}) }[/math], define the series [math]\displaystyle{ \{\psi_{jk}\} }[/math] by
- [math]\displaystyle{ \psi_{jk}(x) = 2^{j/2}\psi(2^jx-k) }[/math]
for integers [math]\displaystyle{ j,k\in \mathbb{Z} }[/math].
Such a function is called an R-function if the linear span of [math]\displaystyle{ \{\psi_{jk}\} }[/math] is dense in [math]\displaystyle{ L^2(\mathbb{R}) }[/math], and if there exist positive constants A, B with [math]\displaystyle{ 0\lt A\leq B \lt \infty }[/math] such that
- [math]\displaystyle{ A \Vert c_{jk} \Vert^2_{l^2} \leq \bigg\Vert \sum_{jk=-\infty}^\infty c_{jk}\psi_{jk}\bigg\Vert^2_{L^2} \leq B \Vert c_{jk} \Vert^2_{l^2}\, }[/math]
for all bi-infinite square summable series [math]\displaystyle{ \{c_{jk}\} }[/math]. Here, [math]\displaystyle{ \Vert \cdot \Vert_{l^2} }[/math] denotes the square-sum norm:
- [math]\displaystyle{ \Vert c_{jk} \Vert^2_{l^2} = \sum_{jk=-\infty}^\infty \vert c_{jk}\vert^2 }[/math]
and [math]\displaystyle{ \Vert \cdot\Vert_{L^2} }[/math] denotes the usual norm on [math]\displaystyle{ L^2(\mathbb{R}) }[/math]:
- [math]\displaystyle{ \Vert f\Vert^2_{L^2}= \int_{-\infty}^\infty \vert f(x)\vert^2 dx }[/math]
By the Riesz representation theorem, there exists a unique dual basis [math]\displaystyle{ \psi^{jk} }[/math] such that
- [math]\displaystyle{ \langle \psi^{jk} \vert \psi_{lm} \rangle = \delta_{jl} \delta_{km} }[/math]
where [math]\displaystyle{ \delta_{jk} }[/math] is the Kronecker delta and [math]\displaystyle{ \langle f \vert g \rangle }[/math] is the usual inner product on [math]\displaystyle{ L^2(\mathbb{R}) }[/math]. Indeed, there exists a unique series representation for a square-integrable function f expressed in this basis:
- [math]\displaystyle{ f(x) = \sum_{jk} \langle \psi^{jk} \vert f \rangle \psi_{jk}(x) }[/math]
If there exists a function [math]\displaystyle{ \tilde{\psi} \in L^2(\mathbb{R}) }[/math] such that
- [math]\displaystyle{ \tilde{\psi}_{jk} = \psi^{jk} }[/math]
then [math]\displaystyle{ \tilde{\psi} }[/math] 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 [math]\displaystyle{ \psi = \tilde{\psi} }[/math], the wavelet is said to be an orthogonal wavelet.
An example of an R-function without a dual is easy to construct. Let [math]\displaystyle{ \phi }[/math] be an orthogonal wavelet. Then define [math]\displaystyle{ \psi(x) = \phi(x) + z\phi(2x) }[/math] for some complex number z. It is straightforward to show that this ψ does not have a wavelet dual.
See also
References
- Charles K. Chui, An Introduction to Wavelets (Wavelet Analysis & Its Applications), (1992), Academic Press, San Diego, ISBN:0-12-174584-8
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