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 [-pi W, pi W] from its sampled values f (k/W), k epsilon Z, in terms of (S-w f)(t) Sigma(infinity)(k=-infinity) f(k/W) sinc(Wt - k) = f (t) (t epsilon R). If f is continuous but not bandlimited, one normally considers lim(W ->infinity)(S-W (f))(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 S-W f to f in the L-p(R)-norm for 1 < p < infinity, the classical modulus of continuity being replaced by the averaged modulus of smoothness tau(r)(f; W-1; M(R))(p). The major theorem enables one to sample any bounded signal f belonging to a certain subspace A(p) of L-p(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 U-norm.