(Generalized multiresolution analysis) We call "generalized multiresolution analysis" (GMRA) the sequence of subspaces , with the following properties.

1.

2. , .

3.

4.

5. such that and is a Riesz basis.

The closures are taken in -norm.

See the definition ( Scale and transport operators 2 ) for and . The function is called the "scaling function".

(GMRA Riesz bases) For each the set is a Riesz basis for . See the definition ( Approximation and detail operators ) for the notation .

Combine definition ( Generalized multiresolution analysis )-4 and formula ( Property of scale and transport 2 ).

(Scaling equation 4) The proposition ( Scaling equation ) extends to GMRA without changes.

See the proof of the proposition ( Scaling equation ).