Classical and approximate sampling theorems; studies in the [formula omitted] and the uniform norm

Abstract

The approximate sampling theorem with its associated aliasing error is due to J.L. 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 ∈ B π w p , 1 ⩽ p < ∞ , 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 ∈ B π w p , 1 < p < ∞ (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 Λ p . Basic in the proof is an intricate result on the representation of the integral ∫ R | f ( u ) | p du as the limit of an infinite Riemann sum of | f | 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.