Definition
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.
Read more about this topic: Dual Wavelet
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