Dual wavelet explained

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(R)} , define the series \{\psi _{jk} \} by\psi _{jk} (x) = 2 ^{j/2} \psi( 2 ^{jx-k)}

for integers

j,k\in Z .

Such a function is called an R-function if the linear span of

\{\psi _{jk} \} is dense in L ^{2(R)} , and if there exist positive constants A, B with 0 <A\leq B < inf ty such thatA \Vert c _{jk} \Vert ^{2} _{l ^{2}} \leq \ Vert \sum ^{inf ty} _{jk =-inf ty} c _{jk} \psi _{jk} \ Vert ^{2} _{L ^{2}} \leq B \Vert c _{jk} \Vert ^{2} _{l ^{2}}

for all bi-infinite square summable series

\{c _{jk} \} . Here, \Vert ⋅ \Vert _{l ^{2}} denotes the square-sum norm:\Vert c _{jk} \Vert ^{2} _{l ^{2}} = \sum ^{inf ty} _{jk =-inf ty} \vert c _{jk} \vert ^{2}

and

\Vert ⋅ \Vert _{L ^{2}} denotes the usual norm on L ^{2(R)} :\Vert f\Vert ^{2} _{L ^{2}} = \int ^{inf ty} _{-inf ty} \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(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(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( 2 x) 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,