Dual wavelet

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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 in general representable by a square integral function itself.

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.

See also

References

  • Charles K. Chui, An Introduction to Wavelets (Wavelet Analysis & Its Applications), (1992), Academic Press, San Diego, ISBN 0-12-174584-8
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