The simplest multiresolution example is the Haar multiresolution. In this case, q is the characteristic function of [0,1], and the basis coefficients in a resolution space are computed by an average on a suitable interval. You can see an example of a Haar multiresolution of classical painting (Pan et Syrinx by Boucher, 1759, National Gallery, London).
The sequence of polynomial splines spaces with steps 2j , j in Z, is a multiresolution approximation.
The following movie (in QuickTime format) shows the succesive approximations of a wavelet classics.