The Exponential Sampling Theorem of Signal Analysis and the Reproducing Kernel Formula in the Mellin Transform Setting

The Shannon sampling theory of signal analysis, the so-called WKS-sampling theorem, which can be established by methods of Fourier analysis, plays an essential role in many fields. The aim of this paper is to study the so-called exponential sampling theorem (ESF) of optical physics and engineering in which the samples are not equally spaced apart as in the Shannon case but exponentially spaced, using the Mellin transform methods. One chief aim of this paper is to study the reproducing kernel formula, not in its Fourier transform setting, but in that of Mellin, for Mellin-bandlimited functions as well as an approximate version for a less restrictive class, the MRKF, namely for functions which are only approximately Mellin-bandlimited. The rate of approximation for such signals will be measured in terms of the strong Mellin fractional differential operators recently studied by the authors. The final aim is to show that the exponential sampling theorem ESF is equivalent to the MRKF for Mellin-bandlimited functions (signals) in the sense that each is a corollary of the other. Three graphical examples are given, as illustration of the theory.