The study of Fredholm equations of the first kind, which are ill-posed inverse problems, is an area of physics and mathematics that is currently flourishing, with applications in many areas of interest. Problems of this type are characterised by a forward relation that includes some loss of information. It is the loss of information that makes calculating the backward relation so difficult. Work is presented here which attempts to add in some of the lost information by making use of such a priori constraints as positivity and known moments. This is achieved by the method of quadratic programming, with a choice of optimisation criteria studied.