On the approximation of nonbandlimited signals by nonuniform sampling series

The classical WKS sampling theorem is a central result in signal processing, but it applies to band-limited signals only. For many purposes, this class of signals is too narrow. For example, the signals that occur in practice are invariably of finite duration, or time-limited, and often have discontinuities. Clearly, such signals cannot be band-limited. We consider the problem of approximating such signals, or other signals not necessarily band-limited, using sampling series. We do not assume that the sampling instants are regularly distributed, in order to account for errors due to jitter. To the best of our knowledge, the problem of obtaining nonuniform sampling approximations for signals not necessarily band-limited, despite its practical interest, has not been addressed in the literature. In this work we introduce a method that leads to sampling approximations with the required properties. It is shown that the sampling sums considered are capable of approximating a wide class of signals, with arbitrarily small L
2
and L
∞
errors.