Dual wavelet
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 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.
