  At measurements of gamma-radiation spectra from ultra-relativistic electrons
in periodic structures, pileup of events in the calorimeter may cause
significant deviation of the detector signal from the classically evaluated
spectrum. That requires appropriate resummation of multiphoton contributions.
We describe the resummation procedure for the photon spectral intensity and for
the photon multiplicity spectrum, and apply it to the study of spectra of
coherent radiation with an admixture of incoherent component. Impact of
multiphoton effects on the shape of the radiation spectrum is investigated. The
limit of high photon multiplicity for coherent radiation is explored. A method
for reconstruction of the underlying single-photon spectrum from the
multiphoton one is proposed.
