Biorthogonal wavelet

Remarks

Under these conditions every $f \in L^2(\R)$ has the expansions $$f = \sum_{j,k \in \Z} \ip{f}{\tilde\psi_{j,k}} \, \psi_{j,k} = \sum_{j,k \in \Z} \ip{f}{\psi_{j,k}} \, \tilde\psi_{j,k},$$ with convergence in $L^2(\R)$.

Example: if $\psi$ is an orthonormal wavelet, such as the Haar wavelet, then $(\psi, \psi)$ is a biorthogonal pair. Each family is an orthonormal basis, hence a Riesz basis, and clause 3 is orthonormality.

Non-example: take $\psi$ the Haar wavelet and $\tilde\psi = 2\psi$. Both families are Riesz bases, but $\ip{\psi_{0,0}}{\tilde\psi_{0,0}} = 2$, so clause 3 fails.

In practice biorthogonal pairs come from two multiresolution analyses with dual scaling functions $\varphi$ and $\tilde\varphi$, and the transform is computed with two pairs of finite filters. The Cohen–Daubechies–Feauveau family contains the 5/3 and 9/7 biorthogonal wavelets used in JPEG 2000.