ON THE SAMPLING THEOREM IN HILBERT SPACES

摘要

Although the sampling theorem is well known, its demonstrationrnassumes generalized delta functions and periodicrnsignals, which are not defined in Hilbert spaces, as well asrnbandlimited signals which not exist in reality because theyrnmust have infinite duration.rnIn this paper, we consider the sampling theorem in Hilbertrnspaces. We present a simple and complete demonstrationrnof the sampling theorem that uses only signals defined inrnHilbert spaces, i.e., we do not use generalized delta functionsrnor periodic signals. We also establish the aliasing effectsrnas a corollary of the theorem.