Generalised sampling theorem for band-limited functions

We present a generalisation of the Whittaker-Shannon sampling theorem for band-limited functions. In the conventional theorem, the set of data from which the function is reconstructed consists of function values that are sampled on a sufficiently fine equidistant grid. In the theorem proved in this paper, the set of data is more general, for example, it may consist of mean values of the function on equidistant intervals. The application of the theorem to accurate determinations of intensities in optics is discussed.