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


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

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^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}\,$$

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


 * $$\Vert c_{jk} \Vert^2_{l^2} = \sum_{jk=-\infty}^\infty \vert c_{jk}\vert^2$$

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


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

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 &psi;. In general, for some given R-function &psi;, 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 &psi; does not have a wavelet dual.