(Scaling equation) Let be an MRA. There exists a sequence such that Furthermore

The equation follows from the proposition ( Subspace bases ) and the definition ( Multiresolution analysis )-1 for . The inclusion follows from the proposition ( Parseval equality ).

We take Fourier transform of : and use the formula ( Property of scale and transport 6 ):

(Scaling filter and auxiliary function) In context of the proposition ( Scaling equation ), the sequence is called "scaling filter" and the function is called "auxiliary function".