Mathematical Problems in Engineering
Volume 2008 (2008), Article ID 164808, 24 pages
doi:10.1155/2008/164808
Research Article
Shannon Wavelets Theory
Department of Pharmaceutical Sciences (DiFarma), University of Salerno, Via Ponte don Melillo, Fisciano, 84084 Salerno, Italy
Received 30 May 2008; Accepted 13 June 2008
Academic Editor: Cristian Toma
Copyright © 2009 Carlo Cattani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Shannon wavelets are studied together with their differential properties (known as connection
coefficients). It is shown that the Shannon sampling theorem can be considered in a more general
approach suitable for analyzing functions ranging in multifrequency bands. This generalization
coincides with the Shannon wavelet reconstruction of L2(ℝ) functions. The differential properties
of Shannon wavelets are also studied through the connection coefficients. It is shown that Shannon
wavelets are C∞-functions and their any order derivatives can be analytically defined by some kind of
a finite hypergeometric series. These coefficients make it possible to define the wavelet reconstruction
of the derivatives of the Cℓ-functions.