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Phase-Retrieval in Shift-Invariant Spaces with Gaussian Generator
Journal of Fourier Analysis and Applications ( IF 1.2 ) Pub Date : 2020-06-15 , DOI: 10.1007/s00041-020-09755-5
Karlheinz Gröchenig

We study the problem of recovering a function of the form \(f(x) = \sum _{k\in \mathbb {Z}} c_k e^{-(x-k)^2}\) from its phaseless samples \(|f(\lambda )|\) on some arbitrary countable set \(\Lambda \subseteq \mathbb {R}\). For real-valued functions this is possible up to a sign for every separated set with Beurling density \(D^-(\Lambda ) >2\). This result is sharp. For complex-valued functions we find all possible solutions with the same phaseless samples.

中文翻译:

高斯发生器在平移不变空间中的相位提取

我们研究了从其无相位样本\(f(x)= \ sum _ {k \ in \ mathbb {Z}} c_k e ^ {-(xk)^ 2} \)中恢复函数的问题。| f(\ lambda)| \)在一些任意可数集\(\ Lambda \ subseteq \ mathbb {R} \)上。对于实值函数,对于Beurling密度\(D ^-(\ Lambda)> 2 \)的每个单独集合,这最多可以有一个符号。这个结果很明显。对于复数值函数,我们找到具有相同无相样本的所有可能解。
更新日期:2020-06-15
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