Wavelet transforms associated with finite cyclic groups

AbstmctMultiresolution 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 so-called Laplacian pyramid schemes, in which the resolution is successively halved by recursively lowpass filtering the signal under analysis and decimating it by a factor of two. Generally speakhg, 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 €mm arbitrary fields. Attention is restricted to sequences of length 2" for n a positive iuteger, so that the resolution may be recursively halved to completion. As in finitelength 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. This development is motivated by potential applications in areas such as digital signal processing and algebraic coding, in which cyclic Fourier analysis has found widespread applications.