Solutions to a filtering problem and convolution equations (Q1260851)

The paper investigates the Tauberian minimization problem which is to characterize the criteria for achieving \(\inf \{I(f)\): \(f\in L^ 1(\mathbb{R})\}\) where \(I(f)\) is the mean square error of the estimation of a function \(y\) by means of \(f*x\) for given functions \(x\) and \(y\). It is shown that the existence of `good' filters \(g\) is equivalent to the existence of solutions of certain kinds of convolution equations. Several necessary conditions are found, and the existence criteria are characterised in terms of the Fourier transform of the observed signal. The results are extended to similar problems in stochastic processes.