Harminv is a free program (and accompanying library) to solve the problem of harmonic inversion — given a discrete-time, finite-length signal that consists of a sum of finitely-many sinusoids (possibly exponentially decaying) in a given bandwidth, it determines the frequencies, decay constants, amplitudes, and phases of those sinusoids.
It can, in principle, provide much better accuracy than straightforwardly extracting FFT peaks, essentially because it assumes a specific form for the signal. (Fourier transforms, in contrast, attempt to represent any data as a sum of sinusoidal components, and are thus limited by the uncertainty principle.) It is also often more robust than directly least-squares fitting the data (which can have problematic convergence), since it re-expresses the problem in terms of simply finding the eigenvalues of a small matrix.
This kind of spectral analysis has wide applications in many areas of physics and engineering, as well as other fields. For example, it could be used to extract the vibrational or "eigen" modes of a system from its response to some stimulus, and also their rates of decay in dissipative systems. FDM has been applied to analyze, e.g., NMR experimental data. It is especially appropriate for analyzing numerical simulations, e.g. of quantum mechanics or classical electromagnetism. In general, it is useful when you know on physical grounds that your system consists of a small number of decaying & oscillating modes in the bandwidth of interest, plus a limited amount of noise, and is not appropriate to analyze completely arbitrary waveforms.