Solutions to a Filtering Problem and Convolution Equations

We investigate the Tauberian minimization problem which is to characterize the criterion for achieving inf{I(f): f ∈ L1(R)} = I(g), g ∈ L1(R). The existence of "good" filters g is equivalent to the existence of solutions to certain kinds of convolution equations. Several necessary conditions are found and the criteria for the existence are characterized in terms of the power spectrum which is the Fourier transform of the observe signal. Further, all the results are extended to similar (but slightly different) problems in stochastic processes.