SIAM Review, Volume 62, Issue 2, Page 299-299, January 2020.
In a previous issue of SIAM Review, I started my introduction to the Survey and Review section by saying that it is safe to bet that most readers use matrices in their work. I would also safely bet that most readers have used the (discrete or continuous) Fourier transform at some point, since, for reasons I have not yet completely understood, Fourier techniques appear in all kinds of unrelated areas of applied mathematics, including many where there is no apparent connection with sinusoidal functions. The paper that follows, “Phase Retrieval: Uniqueness and Stability,” by Philipp Grohs, Sarah Koppensteiner, and Martin Rathmair, studies the problem of recovering a function (f(x)) from the knowledge of the magnitude (|\widehatf(x)|) of its Fourier transform (\widehatf(x)). In applications, including quantum mechanics, astronomy, radar, and speech recognition, this problem has been very relevant for a long time. The most salient application is in the field of diffraction imaging, where one tries to determine an object from diffraction patterns. The study of the structure of crystals by x-ray diffraction got von Laue the 1914 Nobel Prize in Physics, and since then there have been at least another 11 Nobel Prizes related to diffraction imaging. The paper has three sections in addition to the introduction. Sections 3 and 4 are, respectively, devoted to the discrete and continuous Fourier transforms. Section 2 presents a general theory of phase retrieval and successively addresses two questions: (i) to what extent may a function (f) be recovered from the knowledge of the magnitude (|Tf|), where (T) is a linear transformation, and (ii) if the recovery is possible, what happens when (|Tf|) undergoes small perturbations? Since the material has many useful applications and is related to a number of different branches of applied mathematics, it will be of interest to a wide range of readers.

中文翻译：

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调查和审查

SIAM评论，第62卷，第2期，第299-299页，2020年1月。
在上一期《 SIAM评论》中，我从“调查和评论”部分开始介绍，可以肯定地说，大多数读者在他们的工作中使用矩阵。我也可以肯定地说，大多数读者在某个时候都使用过（离散或连续）傅立叶变换，因为由于我尚未完全理解的原因，傅立叶技术出现在应用数学的所有不相关领域，包括许多与正弦函数没有明显联系。紧随其后的是Philipp Grohs，Sarah Koppensteiner和Martin Rathmair撰写的论文“相检索：唯一性和稳定性”，研究了从量级（| \ widehatf（x）的知识中恢复函数（f（x））的问题。 ）|）的傅里叶变换（\ widehatf（x））。在包括量子力学，天文学，雷达，和语音识别，这个问题已经很长时间了。最显着的应用是在衍射成像领域，该领域试图从衍射图样确定物体。通过X射线衍射对晶体结构进行的研究使冯·劳埃（von Laue）获得了1914年诺贝尔物理学奖，此后至少又获得了11项与衍射成像有关的诺贝尔奖。除引言外，本文还包括三个部分。第3节和第4节分别讨论离散和连续傅立叶变换。第2节介绍了相位检索的一般理论，并依次解决了两个问题：（i）从幅度（| Tf |）的知识中可以将函数（f）恢复到什么程度，其中（T）是线性变换， （ii）如果有可能恢复，（| Tf |）受到很小的扰动会怎样？由于该材料具有许多有用的应用程序，并且与应用数学的许多不同分支相关，因此它将引起广泛读者的兴趣。