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 itself in general representable by a square integrable function.
Definition
Given a square integrable function , define the series by
for integers .
Such a function is called an Rfunction if the linear span of is dense in , and if there exist positive constants A, B with such that
for all biinfinite square summable series . Here, denotes the squaresum 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 Rfunction ψ, the dual will not exist. In the special case of , the wavelet is said to be an orthogonal wavelet.
An example of an Rfunction 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
References
 Charles K. Chui, An Introduction to Wavelets (Wavelet Analysis & Its Applications), (1992), Academic Press, San Diego, ISBN 0121745848