Abstract:
The relation of the sampling theorem, developed by Shannon
for Information Theory, to the Whittaker interpolatory functions is
investigated and further extensions in this direction are suggested.
The theory of the Shannon sampling theorem with its application in
Information Theory is reviewed. The Kramer generalization of this
theorem is illustrated for many more functions of interest in Mathematical
Physics, so the use of this theorem in other fields besides
communications is suggested.
Campbell's question concerning the possibility of reducing
Kramer's sampling theorem to that of Whittaker is investigated and
illustrated for more functions of interest. Some useful theorems in
this direction are proposed and proved.
It is shown that although most illustrative functions satisfy
Campbell's conditions, the advantage of Kramer's sampling theorem over that of Whittaker is removed only in the mind of the communications
engineer, who is mainly interested in no integral transforms
other than the celebrated Fourier Transform.
