  This paper discusses the undulator radiation emitted by high-energy positrons
during planar channeling in periodically bent crystals. We demonstrate that the
construction of the undulator for positrons with energies of 10 GeV and above
is only possible if one takes into account the radiative energy losses. The
frequency of the undulator radiation depends on the energy of the particle.
Thus the decrease of the particle's energy during the passage of the crystal
should result in the destruction of the undulator radiation regime. However, we
demonstrate that it is possible to avoid the destructive influence of the
radiative losses on the frequency of the undulator radiation by the appropriate
variation of the shape of the crystal channels. We also discuss a method by
which, to our mind, it would be possible to prepare the crystal with the
desired properties of its channels.
