Classical and approximate sampling theorems;: studies in the LP (R) and the uniform norm

The approximate sampling theorem with its associated aliasing error is due to U. Brown (1957). This theorem includes the classical Whittaker-Kotel'nikov-Shannon theorem as a special case. The converse is established in the present paper, that is, the classical sampling theorem for f is an element of B-pi w(p), 1 <= P < infinity, w > 0, implies the approximate sampling theorem. Consequently, both sampling theorems are fully equivalent in the uniform norm. Turning now to L-p(R)-space, it is shown that the classical sampling theorem for f is an element of B-pi w(p), 1 < p < infinity (here p = 1 must be excluded). implies the L-p(R)-approximate sampling theorem with convergence in the L-P(R)-norm, provided that f is locally Riemann integrable and belongs to a certain class Lambda(p). Basic in the proof is an intricate result on the representation of the integral integral(R) vertical bar f (u)vertical bar(p) du as the limit of an infinite Riemann sum of vertical bar f vertical bar(p) for a general family of partitions of R; it is related to results of O. Shisha et al. (1973-1978) on simply integrable functions and functions of bounded coarse variation on R. These theorems give the missing link between two groups of major equivalent theorems; this will lead to the solution of a conjecture raised a dozen years ago. (c) 2005 Elsevier Inc. All rights reserved.