## 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 $$L_{2}(\mathbb R)$$ 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 Nontrigonometric harmonic analysis involving wavelets and other special systems
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### References:

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