Ill-posed deconvolutions: Regularization and singular value decompositions

We consider 3 procedures that have been proposed and used on the following type of ill-posed problems: Given an experimentally-measured function g, compute f knowing that g(t) = ??0 1 K(t-s)f(s)ds. The 3 procedures are (1) singular value decomposition with truncation; (2) a decomposition procedure of Ekstrom and Rhoads; and (3) the Tikhonov regularization procedure. Relationships between these 3 procedures are discussed with emphasis on the mollifying effects each has on the noise in the measurements. Regularization is shown to provide the most natural setting for noise mollification, although it may not be the best procedure to use.