An Improved Nyquist–Shannon Irregular Sampling Theorem From Local Averages

The Nyquist–Shannon sampling theorem is on the reconstruction of a band-limited signal from its uniformly sampled samples. The higher the signal bandwidth gets, the more challenging the uniform sampling may become. To deal with this problem, signal reconstruction from local averages has been studied in the literature. In this paper, we obtain an improved Nyquist–Shannon sampling theorem from general local averages. In practice, the measurement apparatus gives a weighted average over an asymmetrical interval. As a special case, for local averages from symmetrical interval, we show that the sampling rate is much lower than that of a result by Gröchenig. Moreover, we obtain two exact dual frames from local averages, one of which improves a result by Sun and Zhou. At the end of this paper, as an example application of local average sampling, we consider a reconstruction algorithm: the piecewise linear approximations.