# Multiresolution Approximations Are The Foundation of Dyadic Wavelets

This property allows the implementation of a Fast Dyadic Wavelet Transform with filter banks.

The definition of dyadic wavelets comes from the definition of multiresolution approximations.

This idea of consecutive approximations at finer and finer resolutions is formalized by the concept of multiresolution approximation (or multiresolution analysis).

## Definition

A sequence Vj , j in Z, of subspaces of L2(R) is a multiresolution approximation if the six following conditions are satisfied:

 Condition Interpretation Vj+1 is obtained from Vj by a factor 2 rescaling There exists an underlying dyadic sequence of time grids, e.g., the intervals satisfy a geometric progression with reason 2. For any j, Vj+1 is a subspace of Vj Any low resolution signal is also a high resolution signal. Vj is 2j translation invariant. There is an underlying time grid with step 2j. Condition 1 shows that this grid is obtained from the case j=0 by a 2j rescaling. The intersection of the Vj is 0 in L2. A a zero resolution, the only finite energy signal is 0. The union of the Vj is dense in L2. At the infinite resolution, all finite energy signals are perfectly reproduced. There is a function such that the integer translations of q make a Riesz basis of V0. The resolution Vj is generated by a basis which is obtained by 2j translations of a 2j rescaled q. A Riesz basis is a frame of linearly independent vectors.

A less literary definition is available.

The rescaling of q does not modify the area of its Heisenberg box, but it changes the proportions of the box, like for non dyadic wavelets.

## What of wavelets?

The wavelets are used to build a basis in which are represented the details that are gained between a resolution and the next finer one.