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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 \(L_{2}(\mathbb R)\) 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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\textit{C. Cattani}, Math. Probl. Eng. 2008, Article ID 164808, 24 p. (2008; Zbl 1162.42314)

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