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Dual wavelet

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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 ψ ∈ L 2 ( R ) {\displaystyle \psi \in L^{2}(\mathbb {R} )} , define the series { ψ j k } {\displaystyle \{\psi _{jk}\}} by

ψ j k ( x ) = 2 j / 2 ψ ( 2 j x − k ) {\displaystyle \psi _{jk}(x)=2^{j/2}\psi (2^{j}x-k)}

for integers j , k ∈ Z {\displaystyle j,k\in \mathbb {Z} } .

Such a function is called an R-function if the linear span of { ψ j k } {\displaystyle \{\psi _{jk}\}} is dense in L 2 ( R ) {\displaystyle L^{2}(\mathbb {R} )} , and if there exist positive constants A, B with 0 < A ≤ B < ∞ {\displaystyle 0<A\leq B<\infty } 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 {\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}\,}

for all bi-infinite square summable series { c j k } {\displaystyle \{c_{jk}\}} . Here, ‖ ⋅ ‖ l 2 {\displaystyle \Vert \cdot \Vert _{l^{2}}} denotes the square-sum norm:

‖ c j k ‖ l 2 2 = ∑ j k = − ∞ ∞ | c j k | 2 {\displaystyle \Vert c_{jk}\Vert _{l^{2}}^{2}=\sum _{jk=-\infty }^{\infty }\vert c_{jk}\vert ^{2}}

and ‖ ⋅ ‖ L 2 {\displaystyle \Vert \cdot \Vert _{L^{2}}} denotes the usual norm on L 2 ( R ) {\displaystyle L^{2}(\mathbb {R} )} :

‖ f ‖ L 2 2 = ∫ − ∞ ∞ | f ( x ) | 2 d x {\displaystyle \Vert f\Vert _{L^{2}}^{2}=\int _{-\infty }^{\infty }\vert f(x)\vert ^{2}dx}

By the Riesz representation theorem, there exists a unique dual basis ψ j k {\displaystyle \psi ^{jk}} such that

⟨ ψ j k | ψ l m ⟩ = δ j l δ k m {\displaystyle \langle \psi ^{jk}\vert \psi _{lm}\rangle =\delta _{jl}\delta _{km}}

where δ j k {\displaystyle \delta _{jk}} is the Kronecker delta and ⟨ f | g ⟩ {\displaystyle \langle f\vert g\rangle } is the usual inner product on L 2 ( R ) {\displaystyle L^{2}(\mathbb {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 ) {\displaystyle f(x)=\sum _{jk}\langle \psi ^{jk}\vert f\rangle \psi _{jk}(x)}

If there exists a function ψ ~ ∈ L 2 ( R ) {\displaystyle {\tilde {\psi }}\in L^{2}(\mathbb {R} )} such that

ψ ~ j k = ψ j k {\displaystyle {\tilde {\psi }}_{jk}=\psi ^{jk}}

then ψ ~ {\displaystyle {\tilde {\psi }}} 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 ~ {\displaystyle \psi ={\tilde {\psi }}} , the wavelet is said to be an orthogonal wavelet.

An example of an R-function without a dual is easy to construct. Let ϕ {\displaystyle \phi } be an orthogonal wavelet. Then define ψ ( x ) = ϕ ( x ) + z ϕ ( 2 x ) {\displaystyle \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

References

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