"Shannon sampling theorem: a second look through Mellin analysis"
Bardaro, CarloThe classical (one-dimensional) Shannon sampling theorem states an exact representation of (Fourier) bandlimited functions in terms of an interpolating series, in which the samples are equally spaces apart over the real line. This basic result was deeply studied for many years by Paul Butzer and his collaborators Rudolf Stens, Gerhard Schmeisser and many others. At the end of the eighties, a group of physicists and engineers, obtained a representation of the solution of certain inverse problems arising in optical physics, in terms of an interpolating series in which the samples are exonentially spaced over the positive real line. They used, in a formal way, the Mellin transform. During the nineties, Paul Butzer and his collaborator Stefan Jansche, published several papers in which they obtained a theory of Mellin transform fully independent of Foutier analysis. At the same time, they introduced a mathematically rigorous treatment of the exponential sampling formula for functions with compactly supported Mellin transform. In the present talk we describe the structure of the space of Mellin bandlimited functions. In Fourier analysis, the structure of the Paley-Wiener space of all continuous square integrable functions having compactly supported Fourier transform is precisely described by the classical Paley-Wiener theorem. This basic result characterizes the Paley-Wiener spaces by the Bernstein spaces comprising all the square integrable functions which have an analytic extension to the whole complex plane and are of exponential type (the type being connected with the bandwidth of the Fourier transform). The present lecture is devoted to some our recent extensions of the Paley-Wiener theorem in the Mellin transform setting. In this frame the situation is quite different. Indeed we have shown that a Mellin bandlimited function cannot be extended to the whole complex plane as an entire function, but it can be extended to the Riemann surface of the logarithm as an analytic function. Another interesting and simple approach, which avoids the use of the Riemann surfaces and analytical branches, is obtained by introducing a new concept of analyticity, whcih we have called "polar analyticity". This new approach amounts to taking an analytic function, writing its variable in polar coordinates and treating them as if they were Cartesian coordinates. This enables us to define in a simple way the Mellin-Bernstein spaces, as the counterpart of the Bernstein spaces of Fourier analysis.