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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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References:
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