  The production of final state photons in deep inelastic scattering originates
from photon radiation off leptons or quarks involved in the scattering process.
Photon radiation off quarks involves a contribution from the quark-to-photon
fragmentation function, corresponding to the non-perturbative transition of a
hadronic jet into a single, highly energetic photon accompanied by some limited
hadronic activity. Up to now, this fragmentation function was measured only in
electron-positron annihilation at LEP. We demonstrate by a dedicated
parton-level calculation that a competitive measurement of the quark-to-photon
fragmentation function can be obtained in deep inelastic scattering at HERA.
Such a measurement can be obtained by studying the photon energy spectra in
$\gamma + (0+1)$-jet events, where $\gamma$ denotes a hadronic jet containing a
highly energetic photon (the photon jet). Isolated photons are then defined
from the photon jet by imposing a minimal photon energy fraction. For this
so-called democratic clustering approach, we study the cross sections for
isolated $\gamma + (0+1)$-jet and $\gamma + (1+1)$-jet production as well as
for the inclusive isolated photon production in deep inelastic scattering.
