Classical and approximate sampling theorems; studies in the and the uniform norm |
| |
Authors: | PL Butzer JR Higgins RL Stens |
| |
Institution: | aLehrstuhl A für Mathematik, RWTH Aachen, D-52056 Aachen, Germany;bI.H.P., 4 rue du Bary, 11250 Montclar, France |
| |
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 , 1p<∞, w>0, implies the approximate sampling theorem. Consequently, both sampling theorems are fully equivalent in the uniform norm.Turning now to -space, it is shown that the classical sampling theorem for , 1<p<∞ (here p=1 must be excluded), implies the -approximate sampling theorem with convergence in the -norm, provided that f is locally Riemann integrable and belongs to a certain class Λp. Basic in the proof is an intricate result on the representation of the integral as the limit of an infinite Riemann sum of |f|p for a general family of partitions of ; it is related to results of O. Shisha et al. (1973–1978) on simply integrable functions and functions of bounded coarse variation on . 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. |
| |
Keywords: | Sampling theory Signal theory Approximation by discrete operators |
本文献已被 ScienceDirect 等数据库收录! |
|