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On recovery of functions by sampling series. (English) Zbl 0732.41019

In the classical Shannon sampling theorem Dirichlet’s kernel function sinc(x) is used for reconstituting bandlimited functions by their sampling series. In order to improve the behaviour of convergence of sampling series to non-bandlimited functions other kernel functions have been used in the literature, e.g., de la Vallée-Poussin’s kernel function. In the present paper a special class of kernel functions is used, which includes all kernels of the form \(\phi (x)=\sin c((1+\eta)x)\psi (\eta).\) An upper bound is given for the recovery error (aliasing error) for functions belonging to \(L_ p(R)\cap C(R)\). Moreover a necessary and sufficient condition for f belonging to a certain Besov space is investigated in terms of the generalized sampling series.

MSC:

41A30 Approximation by other special function classes
41A27 Inverse theorems in approximation theory
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