ABSTRACT
In the classical context, the 1D-Whittaker-Kotel’nikov-Shannon theorem is directly applicable to time-dependent signals and is normally formulated in that framework. Multivariate, in the sense of iterated 1D applications of sampling theory permeate many branches of engineering, such as signal analysis, radar as well as laser technology, and many others. These aspects naturally make sampling theory an offspring of 1D-interpolation but do not reflect specifically multi-dimensional constructive approximation. In its customary 1D formulation, known in communication and electrical engineering, the Shannon sampling theorem is related to time-dependent signals and 1D-lattices, so that a condition between a bandwidth and sample rate has to be established. Many of the basic ideas of sampling are drastically generalized in new directions and applied in diverse fields of sampling, far beyond what anyone could have envisioned in the early days of sampling theory.
