A SIMPLE PROOF AND SOME EXTENSIONS OF THE SAMPLING THEOREM

The sampling theorem states essentially that if the frequency spectrum, or Fourier transform, gw of a time function ft vanishes for w outside some interval I , then ft is completely determined by its values at certain discrete sampling points, whose density is proportional to the length of the interval I . This note gives a method of proof of the sampling theorem, both for the case where the interval I is centered at the origin and where it is not, which is somewhat simpler than the previously given proofs, and at the same time is more rigorous, and yields several useful generalizations to functions of several variables and random functions.