The aim of this paper is to present a survey of results concerning the Whittaker-Kotel'nikov-Raabe-Shannon-Someya sampling theorem and its various extensions obtained at Aachen since 1977. This theorem, basic in communication engineering, is often called the cardinal interpolation series theorem in mathematical circles. The interconnections of the sampling theorem (in the setting of Paley-Wiener space) with the theory of Fourier series and integrals are examined. Emphasis is placed upon error analysis, including the aliasing, round-off (or quantization), and time jitter errors. Some new error estimates are given, others are improved; many of the proofs are reduced to a common structure. Both deterministic and probabilistic methods are employed. whereas these results are worked out in detail, the paper also contains a brief discussion of some of the various generalizations. |