[en] Electromagnetic radiation appears as a result of a charged particle movement in free space and also in heterogeneous and non-stationary medium. The radiation spectrum depends on the charged particle motion law, as well as on the law of the medium property chage in space and time. The asymptotics of radiation spectrum, i.e. behaviour of spectral intensity at high frequencies, is studied. It is shown that if a charged particle moves along smooth trajectory or if the change in the medium properties takes place accordng to the law described by a smooth function, the radiation spectrum at high frequencies decreases according to exponential law. Thus, radiation spectrum of a charged particle moving along a smooth trajectory in the medium with gradual heterogeneity and (or) instability is rapidly cut, starting from a certain frequency value. The smooth trajectory means that the charge moves according to the law r = r(t), where vector-function r(t) is continuous with all its derivatives. In much the same way the medium with gradual heterogeneities (or with gradual instability) is described by the functions which are continuous with all their derivatives of any order. The method permitting to determine the upper boundary of radiation spectra is presented