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, we define the ideal string to be that string for which a weighted average of the density is minimum. We prove that the density of this ideal string is not a smooth function. By specializing the optimality criterion, we can also show that the Krein string which is made up by a finite number of concentrated point masses is an ideal string.