Iterative deconvolution for kernels with strictly positive Fourier transforms

In many practical situations, the recovery of information about some phenomenon of interest f reduces to performing deconvolution on indirect measurements g = p ∗ f, corresponding to the convolution of f with a known kernel (point spread function) p. However, in practice, only discrete measurements of g will be available. Consequently, the construction of discrete approximations to f  reduces to deriving discrete versions of g = p ∗ f How this is done depends crucially on what is assumed about the properties of the kernel p. Here, it is assumed that p is symmetric and integrable, and that the Fourier transform p of p is strictly positive. Different discrete versions are obtained depending on how the discretisation of g = p ∗ f is performed. After establishing convergence of the truncated schemes, we discuss the underlying difficulties reflecting their ill-posedness and detail how approximations have to be treated with due care. Some of the technical issues underlying the arguments are treated in the appendix.