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

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 $0<A\leq B<\infty$ such that

A\Vert c_{jk}\Vert _{l^{2}}^{2}\leq {\bigg \Vert }\sum _{jk=-\infty }^{\infty }c_{jk}\psi _{jk}{\bigg \Vert }_{L^{2}}^{2}\leq 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=-\infty }^{\infty }\vert c_{jk}\vert ^{2}

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

\Vert f\Vert _{L^{2}}^{2}=\int _{-\infty }^{\infty }\vert f(x)\vert ^{2}dx

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

\langle \psi ^{jk}\vert \psi _{lm}\rangle =\delta _{jl}\delta _{km}

where $\delta _{jk}$ is the Kronecker delta and $\langle f\vert g\rangle$ 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}\langle \psi ^{jk}\vert f\rangle \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 $\phi$ 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.