In principle, a very bright, monochromatic 1-A\r{} signal with \ensuremath{\Elzxh} \ensuremath{\Delta}\ensuremath{\omega}\ensuremath{\approxeq}${10}^{\mathrm{\ensuremath{-}}8}$--${10}^{\mathrm{\ensuremath{-}}6}$ eV can be filtered from white synchrotron radiation by multiple reflection at grazing incidence from mirrors coated with grazing-incidence antireflection (GIAR) films in which either the films or substrate contain resonant M\"ossbauer nuclei. Typically, nonresonant reflectivities can be suppressed to ${10}^{\mathrm{\ensuremath{-}}4}$--${10}^{\mathrm{\ensuremath{-}}3}$ while maintaining resonant reflectivities of \ensuremath{\approxeq}70%, with half-widths strongly broadened by ``enhancement'' to ${\ensuremath{\Gamma}}_{\mathrm{eff}\mathrm{\ensuremath{\approxeq}}20\mathrm{\ensuremath{\Gamma}}}$. Effective filtering should be possible with two to four reflections, or alternatively, with one to two reflections plus time resolution.By using different combinations of films and substrates, the response can be tailored to give narrow resonance widths \ensuremath{\Delta}\ensuremath{\omega}\ensuremath{\approxeq}\ensuremath{\Gamma} and corresponding delayed scattering times to optimize time filtering, or at the other extreme, to produce broad-width filters with \ensuremath{\Elzxh} \ensuremath{\Delta}\ensuremath{\omega}\ensuremath{\approxeq}100\ensuremath{\Gamma} which would be ideal for a high-resolution x-ray source. In the time response there will be ``quantum beats'' at frequencies ${\ensuremath{\Omega}}_{B}$ due to the interference between the radiation emitted by different hyperfine oscillators, so the beat pattern is determined by the hyperfine splitting. Also, there are two interesting dynamical effects---first, due to the ``enchancement effect'' the coherent decay is speeded up relative to the natural lifetime for incoherent decay and internal conversion absorption; and secondly, there will be ``dynamical beats'' at frequencies ${\ensuremath{\omega}}_{B}$ (superimposed on the quantum-beat spectrum) which is essentially an interference between the natural ``ringing'' of an oscillator at its resonsance frequency ${\ensuremath{\omega}}_{0}$ and the collective response which rings with a median frequency ${\ensuremath{\omega}}_{0}$+${\ensuremath{\omega}}_{B}$.Finally, there is also a multiple-reflection delay to the response, which should be a useful aid for time filtering. This paper develops the general theory for resonant filtering of synchrotron radiation using GIAR films, examining in particular the resulting frequency spectrum, the integrated response, and the time response for resonant $^{57}\mathrm{Fe}$ mirrors coated with \ensuremath{\lambda}/4 GIAR films.