The classical and approximate sampling theorems and their equivalence for entire functions of exponential type

Abstract

It is shown that three versions of the sampling theorem of signal analysis are equivalent in the sense that each can be proved as a corollary of one of the others. The theorems in question are the sampling theorem for functions belonging to the Bernstein space Bσ2, the sampling theorem for functions in Bτ∞, 0<τ<σ, and the approximate sampling theorem for non-bandlimited function. One essential difference to an earlier paper of two of the authors is the avoidance of the deep Paley–Wiener theorem of Fourier analysis.