Stable reconstruction of the T(2) distribution by low-resolution NMR measurements and the classical markov and hausdorf momentum problem

Assuming that an original distribution is a probabilistic measure and the Laplace transforms are known only for a finite number of points that are affected by errors, we develop a method for reconstructing weak-sense mean values obtained by integrating smooth functions with the measure. Our method is useful in NMR if the NMR signal can be represented as a superposition of exponential terms. In these circumstances, we show how the data processing can be related to the classical Hausdorf momentum problem. First, we clarify the meaning of stable spectrum reconstruction, and then develop stable filtering and a mean value reconstruction algorithm. Our method has been tested on both simulated and real sets of spin-spin relaxation curves with noise. In view of this, our method could provide an efficient and accurate reconstruction of spin-spin relaxation data. For any reader interested in applications, a "practical recipe" that is almost self-consistent has been included. Copyright 2000 Academic Press.