In
mathematics, a dual wavelet is the
dual
Dual or Duals may refer to:
Paired/two things
* Dual (mathematics), a notion of paired concepts that mirror one another
** Dual (category theory), a formalization of mathematical duality
*** see more cases in :Duality theories
* Dual (grammatical ...
to a
wavelet
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
. In general, the
wavelet series generated by a
square-integrable function will have a dual series, in the sense of the
Riesz representation theorem
:''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem.''
The Riesz representation theorem, sometimes called the ...
. 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