  We study the phase reconstruction of signals $f$ belonging to complex
Gaussian shift-invariant spaces $V^\infty(\varphi)$ from spectrogram
measurements $|\mathcal{G} f(X)|$ where $\mathcal{G}$ is the Gabor transform
and $X \subseteq \mathbb{R}^2$. An explicit reconstruction formula will
demonstrate that such signals can be recovered from measurements located on
parallel lines in the time-frequency plane by means of a Riesz basis expansion.
Moreover, connectedness assumptions on $|f|$ result in stability estimates in
the situation where one aims to reconstruct $f$ on compact intervals. Driven by
a recent observation that signals in Gaussian shift-invariant spaces are
determined by lattice measurements [Grohs, P., Liehr, L., Injectivity of Gabor
phase retrieval from lattice measurements, Appl. Comput. Harmon. Anal. 62
(2023), pp. 173-193] we prove a sampling result on the stable approximation
from finitely many spectrogram samples. The resulting algorithm provides a
provably stable and convergent approximation technique. In addition, it
constitutes a method of approximating signals in function spaces beyond
$V^\infty(\varphi)$, such as Paley-Wiener spaces.
