In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a dual wavelet is the dual

Dual space

In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...

 to a wavelet

Wavelet

A wavelet is a wave-like oscillation with an amplitude that starts out at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one might see recorded by a seismograph or heart monitor. Generally, wavelets are purposefully crafted to have...

. In general, the wavelet series

Wavelet series

In mathematics, a wavelet series is a representation of a square-integrable function by a certain orthonormal series generated by a wavelet...

 generated by a square integrable function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 will have a dual series, in the sense of the Riesz representation theorem

Riesz representation theorem

There are several well-known theorems in functional analysis known as the Riesz representation theorem. They are named in honour of Frigyes Riesz.- The Hilbert space representation theorem :...

. However, the dual series is not in general representable by a square integral function itself.

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

Dense set

In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...

 in , and if there exist positive constants A, B with such that


for all bi-infinite square summable series . Here, denotes the square-sum norm:


and denotes the usual norm on :


By the Riesz representation theorem

Riesz representation theorem

There are several well-known theorems in functional analysis known as the Riesz representation theorem. They are named in honour of Frigyes Riesz.- The Hilbert space representation theorem :...

, there exists a unique dual basis such that


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:


If there exists a function such that


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

Orthogonal wavelet

An orthogonal wavelet is a wavelet where the associated wavelet transform is orthogonal.That is the inverse wavelet transform is the adjoint of the wavelet transform.If this condition is weakened you may end up with biorthogonal wavelets.- Basics :...

.

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.