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 $\psi \in L^2(\mathbb{ℝ})$, define the series $\{{\psi }_{jk}\}$ by

${\psi }_{jk}(x)=2^{j/2}\psi (2^{j}x-k)$

for integers $j,k\in \mathbb{\mathbb{Z}}$.

Such a function is called an R-function if the linear span of $\{{\psi }_{jk}\}$ is dense in $L^2(\mathbb{ℝ})$, and if there exist positive constants A, B with $0<A\le B<\mathrm{\infty }$ such that

$A\Vert c_{jk}{\Vert }_{l^2}^2\le \Vert \sum _{jk=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_{jk}{\psi }_{jk}{\Vert }_{L^2}^2\le B\Vert c_{jk}{\Vert }_{l^2}^2$

for all bi-infinite square summable series $\{c_{jk}\}$. Here, $\Vert \cdot {\Vert }_{l^2}$ denotes the square-sum norm:

$\Vert c_{jk}{\Vert }_{l^2}^2=\sum _{jk=-\mathrm{\infty }}^{\mathrm{\infty }}|c_{jk}{|}^2$

and $\Vert \cdot {\Vert }_{L^2}$ denotes the usual norm on $L^2(\mathbb{ℝ})$:

$\Vert f{\Vert }_{L^2}^2=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}|f(x){|}^{2}dx$

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

$⟨{\psi }^{jk}|{\psi }_{lm}⟩={\delta }_{jl}{\delta }_{km}$

where ${\delta }_{jk}$ is the Kronecker delta and $⟨f|g⟩$ is the usual inner product on $L^2(\mathbb{ℝ})$. Indeed, there exists a unique series representation for a square-integrable function f expressed in this basis:

$f(x)=\sum _{jk}⟨{\psi }^{jk}|f⟩{\psi }_{jk}(x)$

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

${\tilde{\psi }}_{jk}={\psi }^{jk}$

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

An example of an R-function without a dual is easy to construct. Let $\varphi$ be an orthogonal wavelet. Then define $\psi (x)=\varphi (x)+z\varphi (2x)$ 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