On the reconstruction of signals by a finite number of samples

Abstract

Abstract The reconstruction of signals from a finite number of samples by timelimited kernels is investigated. These timelimited kernels have the advantage, in comparison with the classical bandlimited kernels, that no truncation error occurs. The reconstruction error in dependence on the sampling rate W is shown to be directly connected with the distribution of the zeroes of the kernels Fourier transform. This error is estimated under smoothness conditions posed on the signal. As the major result, it is established that if exactly m samples are used for the reconstruction of a signal, then the degree of approximation is maximally O(W−m) for W → ∞. Some examples of timelimited kernels are constructed with the help of splines, in particular kernels which realize the best possible degree of approximation. It is observed that, using these filters, one obtains very good results for the reconstruction of smooth signals as well as of discontinuous signals.