(Multiresolution analysis) We call "multiresolution analysis" (MRA) the sequence of subspaces , with the following properties.

1.

2. , .

3.

4.

5. such that and

The closures are taken in -norm.

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

(Approximation and detail operators) We introduce the "approximation operators" and "detail operators" according to the formulas

(Subspace bases) Let be an MRA. The set is an orthonormal basis of for every .

Let . Then, according to the definition ( Multiresolution analysis )-4, . Hence, by the definition ( Multiresolution analysis )-4,5, is approximated in with any precision by linear combinations of . But preserves the -norm, see the formula ( Property of scale and transport 2 ). Therefore, is approximated by linear combinations of .

The orthonormality follows from the definition ( Multiresolution analysis )-5 and the formulas ( Property of scale and transport 2 ) and ( Property of scale and transport 3 ).