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] from its sampled values f (k/W), k ∈ Z, in terms of (SWf )(t) ≡ ∞ k=−∞ f k W sinc(Wt −k) = f (t) (t ∈ R). If f is continuous but not bandlimited, one normally considers limW→∞(SWf )(t) in the supremumnorm, 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 SWf to f in the Lp(R)-norm for 1 < p < ∞, the classical modulus of continuity being replaced by the averaged modulus of smoothness τr (f ;W −1;M(R))p. The major theorem enables one to sample any bounded signal f belonging to a certain subspace Λp of Lp(R), the jump discontinuities of which may even form a set of measure zero on R. A corollary gives the counterpart of the approximate sampling theorem, now in the Lp-normThe averaged modulus, so far only studied for functions defined on a compact interval [a, b], had first to be extended to functions defined on the whole real axis R. Basic tools are the de La Vallée Poussin means and a semi-discrete Hilbert transform.  2005 Elsevier Inc. All rights reserved.