The generalized Shannon system in wavelet space

Abstract

The Shannon system is generalized and the expansion of a function in the generalized Shannon system is considered. No study of a wavelet expansion exists without the assumption of 'fast' decay of wavelets. The wavelet psi which is associated with the generalized Shannon system has a 'slow' decay. The expansion of a function in the system is shown to converge at a point which satisfies the Lipschitz condition of order alpha> 0. On the other hand, there is a continuous function whose wavelet expansion in the generalized Shannon system diverges. An observation of Gibbs' phenomenon is also given.