INPUT RECOVERY FROM NOISY OUTPUT DATA, USING REGULARIZED INVERSION OFTHE LAPLACE TRANSFORM

wIn a dynamical system the input is to be recovered from finitely many measurements, blurred by random error, of the output of the system. A s usual, the differential equation describing the system is reduced to multiplication with a polynomial after applying the Laplace transform . It appears that there exists a natural, unbiased, estimator for the Laplace transform of the output, from which an estimator of the input can be obtained by multiplication with the polynomial and subsequent a pplication of a regularized inverse of the Laplace transform. It is po ssible, moreover, to balance the effect of this inverse so that ill-po sedness remains restricted to its actual source: differentiation. The rate of convergence of the integrated mean-square error is a positive power of the number of data. The order of the differential equation ha s an adverse effect on the rate which, on the other hand, increases wi th the smoothness of the input as usual.