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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 systemÂ¿ 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 `solutionÂ¿ 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.