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
for integers
Such a function is called an R-function if the linear span of
for all bi-infinite square summable series
and
By the Riesz representation theorem, there exists a unique dual basis
where
If there exists a function
then
An example of an R-function without a dual is easy to construct. Let