Using the Refinement Equations for the Construction of Pre-wavelets Ii: Powers of Two

We study basic questions of wavelet decompositions associated with multiresolution analysis. A rather complete analysis of multiresolution associated with the solution of a refinement equation is presented. The notion of extensibility of a finite set of Laurent polynomials is shown to be central in the construction of wavelets by decomposition of spaces. Two examples of extensibility, first over the torus and then in complex space minus the coordinate axes are discussed. In each case we are led to a decomposition of the fine space in a multiresolution analysis as a sum of the adjacent coarse space plus an additional space spanned by the multiinteger translates of a finite number of pre-wavelets. Several examples are provided throughout to illustrate the general theory. §