  Ultra-low frequency observations (<100 MHz) are particularly challenging
because they are usually performed in a low signal-to-noise ratio regime due to
the high sky temperature and because of ionospheric disturbances whose effects
are inversely proportional to the observing frequency. Nonetheless, these
observations are crucial to study the emission from low-energy populations of
cosmic rays. We aim to obtain the first thermal-noise limited (~ 1.5 mJy/beam)
deep continuum radio map using the LOFAR Low Band Antenna (LBA) system. Our
demonstration observation targeted the galaxy cluster RX J0603.3+4214 (the
"Toothbrush" cluster). We used the resulting ultra-low frequency (58 MHz) image
to study cosmic-ray acceleration and evolution in the post shock region, as
well as their relation with the presence of a radio halo. We describe the data
reduction we have used to calibrate LOFAR LBA observations. The resulting image
is combined with observations at higher frequencies (LOFAR 150 MHz and VLA 1500
MHz) to extract spectral information. We obtained the first thermal-noise
limited image from an observation carried out with the LOFAR LBA system using
all Dutch stations at a central frequency of 58 MHz. With 8 hours of data, we
reached an rms noise of 1.3 mJy/beam at a resolution of 18" x 11". The
procedure we have developed is an important step forward towards routine
high-fidelity imaging with the LOFAR LBA. The analysis of the radio spectra
shows that the radio relic extends to distances of 800 kpc downstream from the
shock front, larger than what allowed by electron cooling time. Furthermore,
the shock wave started accelerating electrons already at a projected distance
of <300 kpc from the crossing point of the two clusters. These results can be
explained if electrons are reaccelerated downstream by background turbulence
possibly combined with projection effects.
