“…In order for the series to be convergent, it is necessary that the inequality | | < 1 be satisfied (this inequality, as well as geometric series (15), is understood in an operator sense, i.e., as a result of action on a time-dependent function). In Cherenkov beam instabilities, the fastest growing perturbations are those having frequencies close to nk z u [1]; consequently, from the expression for , we can see that the applicability condition for transformation (13) reduces to the obvious inequality (16) The quantities q nd 2k /dt 2k are transformed in an analogous manner. Now, we introduce the operators (17) in terms of which to write Eqs.…”