A powerful technique for the solution of a number of experimental in verse problems, described by an underlying first-kind Fredholm equation is presented. Such problems include, for example, diffraction-limited imaging and the analysis of laser light scattering data. The technique requires the construction of the ‘singular system’ of the propblem which then provides exact orthonormal bases both for the description of sampled and truncated measured data and for the reconstructed continuous object ‘solution’ of the inversion. The singular-system approach may ber regarded as a theory of information which generalises in several directions the well known classical concepts of Shannon and Nyquist.