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 in general representable by a square integral function itself.
Given a square integrable function ψ ∈ L 2 ( R ) , define the series { ψ j k } by
ψ j k ( x ) = 2 j / 2 ψ ( 2 j x − k ) for integers j , k ∈ Z .
Such a function is called an R-function if the linear span of { ψ j k } is dense in L 2 ( R ) , and if there exist positive constants A, B with 0 < A ≤ B < ∞ such that
A ∥ c j k ∥ l 2 2 ≤ ∥ ∑ j k = − ∞ ∞ c j k ψ j k ∥ L 2 2 ≤ B ∥ c j k ∥ l 2 2 for all bi-infinite square summable series { c j k } . Here, ∥ ⋅ ∥ l 2 denotes the square-sum norm:
∥ c j k ∥ l 2 2 = ∑ j k = − ∞ ∞ | c j k | 2 and ∥ ⋅ ∥ L 2 denotes the usual norm on L 2 ( R ) :
∥ f ∥ L 2 2 = ∫ − ∞ ∞ | f ( x ) | 2 d x By the Riesz representation theorem, there exists a unique dual basis ψ j k such that
⟨ ψ j k | ψ l m ⟩ = δ j l δ k m where δ j k is the Kronecker delta and ⟨ f | g ⟩ is the usual inner product on L 2 ( R ) . Indeed, there exists a unique series representation for a square integrable function f expressed in this basis:
f ( x ) = ∑ j k ⟨ ψ j k | f ⟩ ψ j k ( x ) If there exists a function ψ ~ ∈ L 2 ( R ) such that
ψ ~ j k = ψ j k then ψ ~ 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 ψ = ψ ~ , the wavelet is said to be an orthogonal wavelet.
An example of an R-function without a dual is easy to construct. Let ϕ be an orthogonal wavelet. Then define ψ ( x ) = ϕ ( x ) + z ϕ ( 2 x ) for some complex number z. It is straightforward to show that this ψ does not have a wavelet dual.
