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

Abstract

The Whittaker–Shannon–Kotel'nikov sampling theorem enables one to reconstruct signals f bandlimited to [ − π W , π W ] from its sampled values f ( k / W ) , k ∈ Z , in terms of ( S W f ) ( t ) ≡ ∑ k = − ∞ ∞ f ( k W ) sinc ( W t − k ) = f ( t ) ( t ∈ R ) . If f is continuous but not bandlimited, one normally considers lim W → ∞ ( S W f ) ( 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 S W f to f in the L p ( R ) -norm for 1 < p < ∞ , the classical modulus of continuity being replaced by the averaged modulus of smoothness τ r ( f ; W −1 ; M ( R ) ) p . The major theorem enables one to sample any bounded signal f belonging to a certain subspace Λ 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 L p -norm. The 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.