On generalized derivative sampling series expansion

Abstract

Master generalized sampling series expansion is presented for entire functions (signals) coming from a class whose members satisfy an extended exponential boundedness condition. Firstly, estimates are given for the remainder of Maclaurin series of those functions and consequent derivative sampling results are obtained and discussed. The established results are employed in evaluating the related remainder term of signals which occur in sampling series expansion of stochastic processes and random fields (not necessarily stationary or homogeneous) whose spectral kernel satisfies the relaxed exponential boundedness. The derived truncation error upper bounds enable to obtain mean-square master generalized derivative sampling series expansion formulae either for harmonizable Piranashvili-type stochastic processes or for random fields. Finally, being the sampling series convergence rate exponential, almost sure P sampling series expansion formulae are presented.