The bright end of the infrared luminosity functions and the abundance of
  hyperluminous infrared galaxies
  We provide the most accurate estimate yet of the bright end of the infrared
(IR) luminosity functions (LFs) and the abundance of hyperluminous IR galaxies
(HLIRGs) with IR luminosities > 10^13 L_solar, thanks to the combination of the
high sensitivity, angular resolution, and large area of the LOFAR Deep Fields,
which probes an unprecedented dynamic range of luminosity and volume. We
cross-match Herschel sources and LOFAR sources in Bootes (8.63 deg^2), Lockman
Hole (10.28 deg^2), and ELAIS-N1 (6.74 deg^2) with rms sensitivities of around
32, 22, and 20 mJy per beam, respectively. We divide the matched samples into
unique and multiple categories. For the multiple matches, we de-blend the
Herschel fluxes using the LOFAR positions and the 150-MHz flux densities as
priors. We perform spectral energy distribution (SED) fitting, combined with
multi-wavelength counterpart identifications and photometric redshift
estimates, to derive IR luminosities. The depth of the LOFAR data allows us to
identify highly complete (around 92% completeness) samples of bright Herschel
sources with a simple selection based on the 250 micron flux density (45, 40,
and 35 mJy in Bootes, Lockman Hole, and ELAIS-N1, respectively). Most of the
bright Herschel sources fall into the unique category (i.e. a single LOFAR
counterpart). For the multiple matches, there is excellent correspondence
between the radio emission and the far-IR emission. We find a good agreement in
the IR LFs with a previous study out to z around 6 which used de-blended
Herschel data. Our sample gives the strongest and cleanest indication to date
that the population of HLIRGs has surface densities of around 5 to 18 / deg^2
(with variations due to a combination of the applied flux limit and cosmic
variance) and an uncertainty of a factor of 2. In comparison, the GALFORM
semi-analytic model significantly under-predicts the abundance of HLIRGs.
