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The ill-posed problem of restoring object information from finitely many measurements of its spectrum can be solved by using the best approximation in Hilbert spaces appropriately designed to include a priori information about object extent and shape and noise statistics. The procedures that are derived are noniterative, the linear ones extending the minimum-energy band-limited extrapolation methods (and thus related to Gerchberg–Papoulis iteration) and the nonlinear ones generalizing Burg’s maximum-entropy reconstruction of nonnegative objects.