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Shannon wavelets theory. (English) Zbl 1162.42314
Summary: 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 ${L}_{2}\left(ℝ\right)$ functions. The differential properties of Shannon wavelets are also studied through the connection coefficients. It is shown that Shannon wavelets are ${C}^{\infty }$-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 ${C}^{\ell }$-functions.

##### MSC:
 42C40 Wavelets and other special systems