Theory of one-dimensional relativistic oscillator in dispersive medium is developed. It is shown that besides the
threshold there is another special value for the oscillator energy (equals to the double of threshold). When the oscillator
energy gets over this value, the spectrum of the radiated photons number modifies. Within this theory the problem of
radiation by planar channeled positrons, taking into account the medium polarization, is considered. Radiation theory for
planar channeled positrons at arbitrary incidence angle and transverse coordinate is constructed. The spectral
distributions of radiation and energy of radiation are derived. It is shown that both the oscillation amplitude and the
radiation frequency range depend on the initial positron incidence angle and transverse coordinate. Averaging of spectral
distributions over initial transverse coordinates is performed; the formulas for distributions of the number of photons and
energy of radiation for a channeling positron bunch are obtained. It is shown that there is an optimal positron incidence
√2 times smaller than the Lindhard angle.
The form of the spectral distribution of the radiated photons number depends on the medium polarization, which, in turn,
stipulates two special values of the bunch energy: the threshold and the critical one. If the bunch energy is between this
two values, the spectrum of radiated photons has one minimum, otherwise - two minimums.
Theory is compared with experimental results. For the coincidence of the theory with experiment it is necessary to take
into account strong polarization of the atomic electron clouds, through which positrons makes the main contribution in
the radiation pass.