  Phase retrieval refers to the problem of recovering some signal (which is
often modelled as an element of a Hilbert space) from phaseless measurements.
It has been shown that in the deterministic setting phase retrieval from frame
coefficients is always unstable in infinite-dimensional Hilbert spaces [7] and
possibly severely ill-conditioned in finite-dimensional Hilbert spaces [7].
  Recently, it has also been shown that phase retrieval from measurements
induced by the Gabor transform with Gaussian window function is stable under a
more relaxed semi-global phase recovery regime based on atoll functions [1].
  In finite dimensions, we present first evidence that this semi-global
reconstruction regime allows one to do phase retrieval from measurements of
bandlimited signals induced by the discrete Gabor transform in such a way that
the corresponding stability constant only scales like a low order polynomial in
the space dimension. To this end, we utilise reconstruction formulae which have
become common tools in recent years [6,12,18,20].
