WAVELET TRANSFORMS ASSOCIATED WITH FINITE CYCLIC GROUPS

Multiresolution analysis via decomposition on wavelet bases has emerge d as an important tool in the analysis of signals and images when thes e objects are viewed as sequences of complex or real numbers. An impor tant class of multiresolution decompositions are the so-called Laplaci an pyramid schemes, in which the resolution is successively halved by recursively low-pass filtering the signal under analysis and decimatin g it by a factor of two. Generally speaking, the principal framework w ithin which multiresolution techniques have been studied and applied i s the same as that used in the discrete-time Fourier analysis of seque nces of complex numbers. An analogous framework is developed for the m ultiresolution analysis of finite-length sequences of elements from ar bitrary 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 s tructure of the index set of such sequences is exploited to characteri ze 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 w hich cyclic Fourier analysis has found widespread applications.