Approximation error of the Whittaker cardinal series in terms of an averaged modulus of smoothness covering discontinuous signals

The Whittaker–Shannon–Kotel'nikov sampling theorem enables one to reconstruct signals f   bandlimited to [−πW,πW][−πW,πW] from its sampled values f(k/W)f(k/W), k∈Zk∈Z, in terms of(SWf)(t)≡∑k=−∞∞f(kW)sinc(Wt−k)=f(t)(t∈R). If f   is continuous but not bandlimited, one normally considers limW→∞(SWf)(t)limW→∞(SWf)(t) in the supremum-norm, together with aliasing error estimates, expressed in terms of the modulus of continuity of f   or its derivatives. Since in practice signals are however often discontinuous, this paper is concerned with the convergence of SWfSWf to f   in the Lp(R)Lp(R)-norm for 1