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 $\psi\in L^2(\mathbb)$, define the series $\$ by

:$\psi_(x) = 2^\psi(2^jx-k)$

for integers $j,k\in \mathbb$.

Such a function is called an ''R''-function if the linear span of $\$ is dense in $L^2(\mathbb)$, and if there exist positive constants ''A'', ''B'' with 0 such that

:$A \Vert c_ \Vert^2_ \leq \bigg\Vert \sum_^\infty c_\psi_\bigg\Vert^2_ \leq B \Vert c_ \Vert^2_\,$

for all bi-infinite square summable series $\$. Here, $\Vert \cdot \Vert_$ denotes the square-sum norm:

:$\Vert c_ \Vert^2_ = \sum_^\infty \vert c_\vert^2$

and $\Vert \cdot\Vert_$ denotes the usual norm on $L^2(\mathbb)$:

:$\Vert f\Vert^2_= \int_^\infty \vert f(x)\vert^2 dx$

By the Riesz representation theorem, there exists a unique dual basis $\psi^$ such that

:$\langle \psi^ \vert \psi_ \rangle = \delta_ \delta_$

where $\delta_$ is the Kronecker delta and $\langle f \vert g \rangle$ is the usual inner product on $L^2(\mathbb)$. Indeed, there exists a unique series representation for a square-integrable function ''f'' expressed in this basis:

:$f(x) = \sum_ \langle \psi^ \vert f \rangle \psi_(x)$

If there exists a function $\tilde \in L^2(\mathbb)$ such that

:$\tilde_ = \psi^$

then $\tilde$ 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 $\psi = \tilde$, the wavelet is said to be an orthogonal wavelet.

An example of an ''R''-function without a dual is easy to construct. Let $\phi$ be an orthogonal wavelet. Then define $\psi(x) = \phi(x) + z\phi(2x)$ for some complex number ''z''. It is straightforward to show that this ψ does not have a wavelet dual.

See also

* Multiresolution analysis

References

* Charles K. Chui, ''An Introduction to Wavelets (Wavelet Analysis & Its Applications)'', (1992), Academic Press, San Diego, {{ISBN, 0-12-174584-8

Wavelets
Wavelet