Fourier-transform spectroscopy (FTS) is based on the Nyquist-Shannon theorem, which dictates a strict sampling protocol for the measurement of continuous signals. Here we introduce Active Fourier Spectroscopy (AFS), a generalization of FTS in which measurements are still taken in the Fourier-conjugate delay domain, but where the most informative acquisition points are selected adaptively and the spectrum is reconstructed by sequential, uncertainty-aware inference rather than by fixed-grid Fourier inversion. Our approach relies on a theoretical insight that the Nyquist condition applies only under complete ignorance about the system under investigation, and shows how to systematically improve upon it when prior knowledge exists. AFS recovers conventional FTS when the prior is uninformative and falls back to it when prior knowledge is not supported by the data. We demonstrate AFS for optical spectroscopy in three settings vital to research and medicine: clinical FTIR-based blood plasma analysis, optical metrology of structured laser beams, and hyperspectral imaging with an RGB camera. In each case AFS reaches a higher target reconstruction quality from far fewer acquisitions and provides complete uncertainty quantification in real time, propagated rigorously through downstream analysis.