By exact projection in phase space, we derive the generalized Langevin equation (GLE) for convolution-time-filtered observables. We employ a general convolution filter that directly acts on arbitrary phase-space observables and can involve low-pass, high-pass, band-pass, and band-stop components. The derived filter GLE has the same form and properties as the ordinary GLE but exhibits modified potential, mass, and memory friction kernel. Our filter-projection approach has diverse applications and can be used to (i) systematically derive temporally coarse-grained models by low-pass filtering, (ii) undo data smoothing inherent in any experimental measurement process, (iii) decompose data exactly into slow and fast variables that can be analyzed separately and each obey Liouville dynamics. The latter application is suitable for removing slow transient or seasonal (i.e., periodic) components that do not equilibrate over simulation or experimental observation time scales and constitutes an alternative to nonequilibrium approaches. We derive integral formulas for the GLE parameters of filtered data for general systems and filters. For the special case of a Markovian system and a Debye low-pass filter, we derive the filter GLE memory kernel in closed form and show that low-pass smoothing of Markovian data induces exponentially decaying memory.
