Wavelets associated with periodic basis functions

In this paper, we investigate a class of nonstationary, orthogonal periodic scaling functions and wavelets generated by continuously differentiable periodic functions with positive Fourier coefficients; such functions are termed periodic basis functions. For this class of wavelets, the decomposition and reconstruction coefficients can be computed in terms of the discrete Fourier transform, so that FFT methods apply for their evaluation. In addition, decomposition at the nth level only involves 2 terms from the higher level. Similar remarks apply for reconstruction. We apply a periodic ''uncertainty principle'' to obtain an angle/frequency uncertainty ''window'' for these wavelets, and we show that for many wavelets in this class the angle/frequency localization is good. (C) 1996 Academic Press, Inc.