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.
Given a square-integrable function
Such a function is called an R-function if the linear span of
, and if there exist positive constants A, B with
such that for all bi-infinite square summable series
denotes the square-sum norm: and
denotes the usual norm on
: By the Riesz representation theorem, there exists a unique dual basis
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
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
An example of an R-function without a dual is easy to construct.
be an orthogonal wavelet.
for some complex number z.
It is straightforward to show that this ψ does not have a wavelet dual.