Wavelets via their fourier transforms

Abstract

As the most simple examples of wavelets and scaling functions which are expressed in terms of their Fourier transforms, we construct and study the generalized Shannon wavelets (G-Shannon wavelets) and the generalized Shannon scaling functions (G-Shannon scaling functions) whose Fourier transforms are given by characteristic functions. One of the features of the G-Shannon wavelets is that they may or may not be associated with MRA. We characterize those G-Shannon wavelets which can be associated with MRA and give a criterion to determine whether a wavelet from a class of G-Shannon wavelets of Ha et al. can be associated with MRA or not. Another feature of the G-Shannon wavelets is the convergence of a G-Shannon wavelet expansion influenced by the slow decay of the G-Shannon wavelets. We study the pointwise convergence and the Gibbs phenomenon on the G-Shannon wavelet expansions. In contrast to the regular wavelet expansion, there is a continuous function whose G-Shannon wavelet expansion diverges. We also see that the G-Shannon wavelet is a sampling function and has the corresponding sampling theorem. By the smoothing procedure of Meyer, the generalized Meyer wavelet is constructed from the G-Shannon wavelet which has a fast decay and satisfies an oversampling theorem.