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

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

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

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

${\displaystyle A\Vert c_{jk}\Vert _{l^{2}}^{2}\leq {\bigg \Vert }\sum _{jk=-\infty }^{\infty }c_{jk}\psi _{jk}{\bigg \Vert }_{L^{2}}^{2}\leq B\Vert c_{jk}\Vert _{l^{2}}^{2}\,}$

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

${\displaystyle \Vert c_{jk}\Vert _{l^{2}}^{2}=\sum _{jk=-\infty }^{\infty }\vert c_{jk}\vert ^{2}}$

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

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

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

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

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

${\displaystyle f(x)=\sum _{jk}\langle \psi ^{jk}\vert f\rangle \psi _{jk}(x)}$

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

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

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

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