"The exponential sampling theorem, related results and its generalizations"
Illaria, MantelliniThe exponential sampling theorem for Mellin badlimited functions was first introduced, in a formal way, by a group of physicists and engineers during the eighties, with the aim to study certain problems arising in optical physics. However a rigorous treatment was given by Paul Butzer and his collaborator Stephan Jansche during the second half of the nineties, involving the theory of Mellin transform. The exponential samplin theorem can be viewed as the Mellin counterpart of the Shannon sampling theorem of Fourier analysis and these formulations are formally equivalent using suitable changes of variables. However this equivalence is only formal. Indeed, in both the results the notions of Fourier bandlimited and Mellin bandlimited function are fundamental. In recent papers we have shown that these classes are really different: the first is linked to the classical Bernstein spaces, the second one to the Mellin-Bernstein spaces. These spaces are completely different. Therefore a deep study of the exponential sampling independent of the classical Shannon sampling appears of interest. In classical Shannon sampling theory, recent results about equivalent formulations of the Shannon sampling formula in terms of other basic formulae of Fourier analysis, like e.g. the reproducing kernel formula, the Parseval formula, were obtained by Paul Butzer, Rudolf Stens, Gerhard Schmeisser and others. Also, they established approximate version of the above results, when the involved function is not bandlimited: under certain basic assumptions, in this case they obtain a formula with a remainder which was estimated in a precise way using a suitable notion of distance based on the Fourier inversion formula. The present talk is devoted to the extensions of these results to the exponential sampling, also in its approximate version. In collaboration with Paul Butzer and Gerhard Schmeisser we studied first the equivalence of the exponential sampling formula, the Mellin reproducing kernel formula and the Mellin-Parseval formula, and then we considered their approximate versions, obtaining precise estimates of the remainder in terms of a new notion of "Mellin distance" defined through the Mellin inversion. Finally, we have recently introduced a generalization of the exponential sampling in which the kernel is replaced by a function satisfying suitable assumptions, which enables one to obtain approximate version of the exponential formula for not necessarily Mellin bandlimited functions. These generalized series represent the Mellin counterpart of the generalized sampling series, introduced, in a general and rigourous form, by Paul Butzer, Rudolf Stens, Gerhard Schmeisser and others, during the eighties.