Fast Pade transform in the theory of resonances: exact solution of the harmonic inversion problem

We present the fast Pade transform (FPT) as a polynomial quotient for the Green or response function from signal processing, spectroscopy and resonant scattering theory. Specific illustrations are given for nuclear magnetic resonance spectroscopy within the problem of harmonic inversion, quantification or spectral analysis. Here, the input time signal points or auto-correlation functions are given via measurements or computations, and the task is to reconstruct the unknown components as the harmonic variables in terms of the fundamental complex frequencies and amplitudes. The FPT solves the harmonic inverse problem exactly by retrieving the true number of resonances with all their proper spectral parameters. This output list is finalized by means of Froissart doublets or pole-zero confluences for unequivocal disentangling of the physical/genuine from unphysical/spurious contents of the analysed time signal. Stability of investigated systems under external perturbations is especially challenged by the presence of noise. The FPT can assess the system's stability through determining the locations and distributions of spectral poles and zeros in the complex frequency plane. This permits identification of the regions that are void of noise, and gives the possibility for improved system performance under more stable conditions with full signal-noise separation. The FPT can provide a number of important biophysical and chemical quantities, including the density of states and abundance or concentrations of all the physical constituents from the investigated substance.