Sampling theory for not necessarily band-limited functions: A historical overview. (English) Zbl 0746.94002

Butzer, P. L.; Stens, R. L.

SIAM Rev. 34, No. 1, 40-53 (1992).

Summary: Shannon’s sampling theorem is one of the most powerful results in signal analysis. The aim of this overview is to show that one of its roots is a basic paper of de la Vallée Poussin of 1908. The historical development of sampling theory from 1908 to the present, especially the matter dealing with not necessarily band-limited functions (which includes the duration-limited case actually studied in 1908), is sketched. Emphasis is put on the study of error estimates, as well as on the delicate pointwise behavior of sampling sums at discontinuity points of the signal to be reconstructed.

MSC:

94A05	Communication theory
94A15	Information theory (general)
41A05	Interpolation in approximation theory
41A45	Approximation by arbitrary linear expressions
41A25	Rate of convergence, degree of approximation
94-03	History of information and communication theory

Keywords:

communication and information theory; interpolation; approximation by linear expressions; degree of approximation; Shannon’s sampling theorem; signal analysis; band-limited functions