Linear inverse problems with discrete data. I. General formulation and singular system analysis

This paper is the first part of a vtᵒrk which is concerned with linear methods for the solution of linear inverse problems with discrte data. Such problems occur frequently in instrumental science, for example tomography, radar, sonar, optical imaging, particle sizing and so on. We give a gnerai formulation of the problem by extending the approach of Backus and Gilbert and by defining a mapping from an infinite-dimensional function space into a finite-dimensional vector space. The singular System of this mapping is introduced and used to define natural bases both in the solution and in the data space. We analyse in this context normal solutions, least-squares solutions and generalised inverses. We illustrate the wide applicability of the singular system technique by discussing several examples in dtail. P articular attention is devoted to showing the many connections between this method and techniques developed in other topics like the extrapolation of band-limited signais and the interpolation of functions specified on a finite set of points. For example, orthogonal polynomials for least-squares approximation, spline functions and discrte prolate spheroidal functions are particular cases of the singular functions introduced in this paper.