HOME

TheInfoList



OR:

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 \psi\in L^2(\mathbb), define the series \ by :\psi_(x) = 2^\psi(2^jx-k) for integers j,k\in \mathbb. Such a function is called an ''R''-function if the linear span of \ is dense in L^2(\mathbb), and if there exist positive constants ''A'', ''B'' with 0 such that :A \Vert c_ \Vert^2_ \leq \bigg\Vert \sum_^\infty c_\psi_\bigg\Vert^2_ \leq B \Vert c_ \Vert^2_\, for all bi-infinite square summable series \. Here, \Vert \cdot \Vert_ denotes the square-sum norm: :\Vert c_ \Vert^2_ = \sum_^\infty \vert c_\vert^2 and \Vert \cdot\Vert_ denotes the usual norm on L^2(\mathbb): :\Vert f\Vert^2_= \int_^\infty \vert f(x)\vert^2 dx By 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 ...
, there exists a unique dual basis \psi^ such that :\langle \psi^ \vert \psi_ \rangle = \delta_ \delta_ where \delta_ is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 ...
and \langle f \vert g \rangle is the usual
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
on L^2(\mathbb). Indeed, there exists a unique series representation for a square-integrable function ''f'' expressed in this basis: :f(x) = \sum_ \langle \psi^ \vert f \rangle \psi_(x) If there exists a function \tilde \in L^2(\mathbb) such that :\tilde_ = \psi^ then \tilde 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 \psi = \tilde, the wavelet is said to be an orthogonal wavelet. An example of an ''R''-function without a dual is easy to construct. Let \phi be an orthogonal wavelet. Then define \psi(x) = \phi(x) + z\phi(2x) for some complex number ''z''. It is straightforward to show that this ψ does not have a wavelet dual.


See also

*
Multiresolution analysis A multiresolution analysis (MRA) or multiscale approximation (MSA) is the design method of most of the practically relevant discrete wavelet transforms (DWT) and the justification for the algorithm of the fast wavelet transform (FWT). It was intr ...


References

* Charles K. Chui, ''An Introduction to Wavelets (Wavelet Analysis & Its Applications)'', (1992), Academic Press, San Diego, {{ISBN, 0-12-174584-8 Wavelets
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 ...