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This interdisciplinary book presents a joint framework with the unified quantum-mechanical theories of resonant scattering, spectroscopy, and signal processing. It deals with S- and R-matrices, variational principles, complex coordinate scaling, wave packet propagation, Fredholm determinants, finite-rank separable expansions, filter diagonalisation, the Lanczos algorithm, and the Padé methodology. Being both mathematically and physically rigorous, the book can be successfully used in optimally quantifying resonances in physics, chemistry, biology, and medical diagnostics as well as in the applied area of signal processing.