Two Remarks on the Reconstruction of Sampled Non-Bandlimited Functions

Abstract

This communication considers the error in the reconstruction of a deterministic, non-bandlimited real function from its sampled values. We have two objectives in mind. First, we show that in a certain sense, to be made precise in the sequel, the error in the reconstruction of f(t) from its sampled values f(nπ/Ω) by sin x/x interpolation is small provided the portion of the amplitude spectrum lying outside [−Ω, Ω]i s small. Our second objective is tos how the analogous situation need not hold for the energy spectrum. Indeed, we shall exhibit a function with an arbitrarily small amount of its energy outside the frequency band |w| ≤ Ω for which the sin x/x interpolation series of equation fails to converge either pointwise or in the mean square sense.