A different perspective: Hermite-type exponential sampling series

When we look at the classical rather than the exponential sampling operators, these operators are modelled on the sampling expansion for bandlimited functions given by the Whittaker-Kotel’nikov-Shannon theorem. Some variations of this classical theorem have been proposed in many works. One of them (going back to Jagerman and Fogel and, more generally, to Linden and Abramson) also considers derivative instances for the reconstruction of bandlimited functions and consequently provides the advantage of a larger sampling rate compared to the Whittaker-Kotel’nikov-Shannon theorem. Very recently, a modification of generalized sampling operators similar to this paper has been considered by R. Corso. Taking this paper into account, we modify the exponential sampling operators to include sampling of Mellin derivatives up to a general order to approximate Mellin-bandlimited functions which need not be necessary. We investigate the basic approximation properties and rate of convergence of the series which we call Hermite-type exponential sampling series. Finally, we present numerical results and graphical representations comparing the new operator with the classical one, considering some examples of kernels that support our main results.