Hyperspectral imaging systems based on multiple-beam interference (MBI), such as Fabry-Perot interferometry, are attracting interest due to its compact design, better throughput, and finer resolution. Unlike dispersive devices, spectra in interferometric systems are reconstructed from raw interferograms. Although the response function of MBI devices is modeled by the Airy function, existing reconstruction techniques are often constrained to the Fourier-transform spectroscopy, which is tailored for two-beam interference (TBI). Those approaches pose limitations for MBI and are susceptible to non-idealities like irregular sampling and noise. To address these challenges, the instrumental response can instead be represented with a transfer matrix. In this paper, we establish a unified formulation of both MBI and TBI spectroscopy, smoothly transitioning from the classical physical interpretation to a numerical system analysis of this forward matrix representation. Consequently, we extend the range of existing techniques for spectrum reconstruction, framing them as a classic optimization problem. Specifically, we employ a Bayesian framework incorporating prior knowledge, such as sparsity constraints. Experiments on simulated and real data demonstrate the framework’s flexibility and noise robustness compared to conventional algorithms. An implementation of the code is available at https://github.com/mhmdjouni/inverspyctrometry.