Ideal solution of an inverse normal mode problem with finite spectral data

The problem of reconstructing the density of a vibrating string given the first N eigenfrequencies for two vibrating configurations admits an infinite number of solutions. Among all such strings compatible with the truncated data set, the ideal string was defined to be that string for which a weighted average of the density is minimum. It was proven that this ideal string must have a finite number of degrees of freedom and hence, that it is made up by a finite number of concentrated point masses. By specializing the optimality criterion, it was shown that the Krein string is an ideal string.