Biorthogonal
Multiresolution Approximations

A pair [(V_j),(V^*_l)] of multiresolution approximations is a biorthogonal multiresolution system if and only if

Then V^*₀ has a Riesz basis of the form q^*(t-n), n in Z, such that the translations of q and of q^* form a biorthogonal system:

We have a biorthogonal bases system instead of a single orthogonal basis.

Example

Below is a pair of biorthogonal scaling functions. The first one is a cubic B-spline. Since there is no orthogonality constraint, the B-spline can be kept as an atom et preserve thus both symmetry and compact support.

Back to the Properties of Multiresolution Approximations