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
![{\displaystyle \psi _{jk}(x)=2^{j/2}\psi (2^{j}x-k)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
![{\displaystyle A\Vert c_{jk}\Vert _{l^{2}}^{2}\leq {\bigg \Vert }\sum _{jk=-\infty }^{\infty }c_{jk}\psi _{jk}{\bigg \Vert }_{L^{2}}^{2}\leq B\Vert c_{jk}\Vert _{l^{2}}^{2}\,}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
for all bi-infinite square summable series
. Here,
denotes the square-sum norm:
![{\displaystyle \Vert c_{jk}\Vert _{l^{2}}^{2}=\sum _{jk=-\infty }^{\infty }\vert c_{jk}\vert ^{2}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
and
denotes the usual norm on
:
![{\displaystyle \Vert f\Vert _{L^{2}}^{2}=\int _{-\infty }^{\infty }\vert f(x)\vert ^{2}dx}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
By the Riesz representation theorem, there exists a unique dual basis
such that
![{\displaystyle \langle \psi ^{jk}\vert \psi _{lm}\rangle =\delta _{jl}\delta _{km}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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:
![{\displaystyle f(x)=\sum _{jk}\langle \psi ^{jk}\vert f\rangle \psi _{jk}(x)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
If there exists a function
such that
![{\displaystyle {\tilde {\psi }}_{jk}=\psi ^{jk}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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.
See also
References
- Charles K. Chui, An Introduction to Wavelets (Wavelet Analysis & Its Applications), (1992), Academic Press, San Diego, ISBN 0-12-174584-8