Discrete data fourier deconvolution

In many practical situations, the recovery of information about some phenomenon of interest f reduces to performing Fourier deconvolution on indirect measurements g = p*f, corresponding to the Fourier convolution of f with a known kernel (point spread function) p. An iterative procedure is proposed for performing the deconvolution of g = p * f, which generates the partial sums of a Neumann series. However, the standard convergence analysis for the Neumann series is not applicable for such deconvolutions so a proof is given which is based on using Fourier properties in L². In practice, only discrete measurements (g_m) of g will be available. Consequently, the construction of a discrete approximation (f_m) to f reduces to performing a deconvolution using a discrete version (g_m) = (p_m) * (f_m) of g = p * f. For p(x) = sech(x)/Π, it is shown computationally, using the discrete version of the proposed iteration, that the resulting accuracy of (f_m) will depend on the form and smoothness of f, the size of the interval truncation, and the level of discretization of themeasurements (g_m). Excellent accuracy for (f_m) is obtainedwhen fgmg and f pmg accurately approximate the essential structure in g and p, respectively, the support of p is much smaller than that for g, and the discrete measurements of (g_m) are on a suitably fine grid.