Wavelet transforms associated with finite cyclic groups

Multiresolution analysis via decomposition on wavelet bases has emerged as an important tool in the analysis of signals and images when these objects are viewed as sequences of complex or real numbers. An important class of multiresolution decompositions are the Laplacian pyramid schemes, in which the resolution is successively halved by recursively low-pass filtering the signal under analysis and decimating it by a factor of two. In general, the principal framework within which multiresolution techniques have been studied and applied is the same as that used in the discrete-time Fourier analysis of sequences of complex numbers. An analogous framework is developed for the multiresolution analysis of finite-length sequences of elements from arbitrary fields. Attention is restricted to sequences of length 2ⁿ, for n a positive integer, so that the resolution may be recursively halved to completion. As in finite-length Fourier analysis, a cyclic group structure of the index set of such sequences is exploited to characterize the transforms of interest for the particular cases of complex and finite fields