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

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Given a square-integrable function  , define the series   by

 

for integers  .

Such a function is called an R-function if the linear span of   is dense in  , and if there exist positive constants A, B with   such that

 

for all bi-infinite square summable series  . Here,   denotes the square-sum norm:

 

and   denotes the usual norm on  :

 

By the Riesz representation theorem, there exists a unique dual basis   such that

 

where   is the Kronecker delta and   is the usual inner product on  . Indeed, there exists a unique series representation for a square-integrable function f expressed in this basis:

 

If there exists a function   such that

 

then   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  , the wavelet is said to be an orthogonal wavelet.

An example of an R-function without a dual is easy to construct. Let   be an orthogonal wavelet. Then define   for some complex number z. It is straightforward to show that this ψ does not have a wavelet dual.

See also

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References

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