There are several reasons why the classical sampling theorem is rather impractical for real life signal processing. First, the sinc-kernel is not very suitable for fast and efficient computation; it decays much too slowly. Second, in practice only a finite number N of sampled values are available, so that the representation of a signal f by the finite sum would entail a truncation error which decreases rather slowly for N¿ ¿, due to the first drawback. Third, band-limitation is a definite restriction, due to the nonconformity of band and time-limited signals. Further, the samples needed extend from the entire past to the full future, relative to some time t = t₀. This paper presents an approach to overcome these difficulties. The sinc-function is replaced by certain simple linear combinations of shifted B-splines, only a finite number of samples from the past need be available. This deterministic approach can be used to process arbitrary, not necessarily bandlimited nor differentiable signals, and even not necessarily continuous signals. Best possible error estimates in terms of an L^p-average modulus of smoothness are presented. Several typical examples exhibiting the various problems involved are worked out in detail.

In this article, we study a nonlinear version of the sampling Kantorovich type operators in a multivariate setting and we show applications to image processing. By means of the above operators, we are able to reconstruct continuous and uniformly continuous signals/images (functions). Moreover, we study the modular convergence of these operators in the setting of Orlicz spaces L ^ϕ(ℝ ⁿ ) that allows us to deal the case of not necessarily continuous signals/images. The convergence theorems in L ^p (ℝ ⁿ )-spaces, L ^αlog ^β L(ℝ ⁿ )-spaces and exponential spaces follow as particular cases. Several graphical representations, for the various examples and image processing applications are included.