Wavelet frames of dilation factor M

Abstract

Based on a frame multiresolution analysis (FMRA) of dilation factor M, we establish quantitative criteria in the frequency domain for the construction of a tight wavelet frame generated by a minimum number of M-1 mother wavelets. We utilize the concept of fibres of closed shift-invariant subspaces in the frequency domain to show the existence of tight frames in the wavelet spaces and also to provide a method of constructing the mother wavelets. It is shown that M mother wavelets would always be sufficient to generate a tight wavelet frame. We also provide an alternative way of constructing the mother wavelets for the elementary case of M=2 based on the analysis of the Gramian. Finally, we extend our construction techniques to creating a sequence space FMRA which is related to periodic wavelet frames and show that the fibres of multiresolution subspaces in L²(R) also form a FMRA in l²(Z) and that the wavelets constructed for L²(R) can be used to construct wavelets for l²(Z).