Publication

Abstract : Gabardo and Nashed [‘Nonuniform multiresolution analyses and spectral pairs’, J. Funct. Anal. 158(1) (1998), 209–241] have introduced the concept of nonuniform multiresolution analysis (NUMRA), based on the theory of spectral pairs, in which the associated translated set Λ={0,r/N}+2Z is not necessarily a discrete subgroup of R , and the translation factor is 2N . Here r is an odd integer with 1≤r≤2N−1 such that r and N are relatively prime. The nonuniform wavelets associated with NUMRA can be used in signal processing, sampling theory, speech recognition and various other areas, where instead of integer shifts nonuniform shifts are needed. In order to further generalize this useful NUMRA, we consider the set Λ˜={0,r1/N,r2/N,…,rq/N}+sZ , where s is an even integer, q∈N , ri is an integer such that 1≤ri≤sN−1,(ri,N)=1 for all i and N≥2 . In this paper, we prove that the concept of NUMRA with the translation set Λ˜ is possible only if Λ˜ is of the form {0,r/N}+sZ . Next we introduce Λs -nonuniform multiresolution analysis ( Λs -NUMRA) for which the translation set is Λs={0,r/N}+sZ and the dilation factor is sN , where s is an even integer. Also, we characterize the scaling functions associated with Λs -NUMRA and we give necessary and sufficient conditions for wavelet filters associated with Λs -NUMRA.