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This paper considers the possible amplification in dielectric media of x -ray and vacuum ultraviolet radiation by electron beams.The medium can be a uniform dielectric resulting in stimulated Cherenkov radiation or a periodic, inhomogenious dielectric resulting in stimulated transition radiation. Two distinct geometries for resonators are considered.In one case, a simple confocal resonator is placed at an obliqueangle to the electron beam.In the other, a conical mirror is used to reflect the entire radiation cone back onto the electron beam. The small signal gain for a short interaction length is found for both these cases to vary linearly with frequency resulting in higher gain for shorter wavelengths.