Orthonormal wavelets for finite sequences

The author deals with the problems encountered when wavelet transforms of finite sequences are considered. It is shown that there are two main drawbacks. First, there is no conservation of the supports. This means, in particular, that the transform generates more coefficients than samples in the signal. Second, the statistical properties of the coefficients are not homogeneous, because of boundary effects. Two methods are proposed to construct bases devoted to finite sequences. The first one is based on a periodization of the signal and makes use of a periodic multiresolution analysis. The second one introduces further symmetrizations in order to obtain a smoother extrapolation of the signal