On the effective bandwidth of sampled signals

Shannon´s Sampling Theorem states that if the sampling frequency of a band limited function is chosen to be larger than twice the cutoff frequency of the signal then the cardinal series obtained from the samples coincides with the original function. Hence, we consider signals that are not band limited. In this case there will be an error between the cardinal series and the original function, no matter how fast one samples. In this paper, we show that for signals which can be expressed as an inverse Lebesgue-Stieltjes transform of a function of bounded variation, the cardinal series provides an arbitrarily close uniform approximation by choosing the sampling frequency sufficiently large. We also obtain a lower bound on how small a sampling frequency can be tolerated based on an error criterion.