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.

翻译：我们研究属于复高斯平移不变空间 $V^\infty(\varphi)$ 的信号 $f$ 从谱图测量 $|\mathcal{G} f(X)|$ 中的相位重构问题，其中 $\mathcal{G}$ 为Gabor变换，$X \subseteq \mathbb{R}^2$。一个显式重构公式将表明，这类信号可通过Riesz基展开，从时频平面上平行线位置的测量值中恢复。此外，关于 $|f|$ 的连通性假设可在旨在紧凑区间上重构 $f$ 的情形下导出稳定性估计。受近期关于高斯平移不变空间中信号由晶格测量唯一确定这一发现[Grohs, P., Liehr, L., Injectivity of Gabor phase retrieval from lattice measurements, Appl. Comput. Harmon. Anal. 62 (2023), pp. 173-193]的驱动，我们证明了从有限个谱图样本进行稳定逼近的采样定理。由此产生的算法提供了一种可证明稳定且收敛的逼近技术。此外，该方法还可用于逼近 $V^\infty(\varphi)$ 之外函数空间中的信号，例如Paley-Wiener空间。