Classical and approximate sampling theorems; studies in the LP(R) 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 p w , 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 p w , 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 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. (1973Shisha et al. ( -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.

Обобщение Теоремы Отсчетов Уиттекера - Котельникова - Шеннона Для Непрерывных Функций На Отрезке

“…Ими установлено, что для равномерно непрерывных ограниченных на R функций f , принадлежащих классу Дини-Липшица {f : lim δ→+0 ω(f, δ) ln 1 δ = 0}, где ω(f, δ) -модуль непрерывности функции f , и, кроме того, удовлетворяющих условию f (x) = O(|x| −δ ) при x → ±∞ для некоторого δ > 0, равномерно на R для любых m ∈ N и 0 < a < 1 справедливо равенство Кроме того, для этих аппроксимаций авторами работы [28] получены теоремы типа Джексона при W → ∞. Интересный признак равномерной сходимости на оси самих кардинальных функций Уиттекера приводится в [29]. Для функций из класса…”