This study investigates the phase retrieval problem for wide-band signals. We solve the following problem: given \(f\in L^2(\mathbb {R})\) with Fourier transform in \(L^2(\mathbb {R},e^{2c|x|}\,\text{ d }x)\), we find all functions \(g\in L^2(\mathbb {R})\) with Fourier transform in \(L^2(\mathbb {R},e^{2c|x|}\,\text{ d }x)\), such that \(|f(x)|=|g(x)|\) for all \(x\in \mathbb {R}\). To do so, we first translate the problem to functions in the Hardy spaces on the disc via a conformal bijection, and take advantage of the inner-outer factorization. We also consider the same problem with additional constraints involving some transforms of f and g, and determine if these constraints force uniqueness of the solution.

中文翻译：

---

宽带信号的相位检索

这项研究调查宽带信号的相位检索问题。我们解决了以下问题：给\（f \ in L ^ 2（\ mathbb {R}）\）在\（L ^ 2（\ mathbb {R}，e ^ {2c | x |} \}中进行傅立叶变换， \ text {d} x）\），我们找到所有函数\（g \ in L ^ 2（\ mathbb {R}）\）在\（L ^ 2（\ mathbb {R}，e ^ { 2c | x |} \，\ text {d} x）\），这样对于所有\（x \ in \ mathbb {R} \）来说\（| f（x）| = | g（x）| \）。为此，我们首先通过共形双射将问题转化为光盘上的Hardy空间中的函数，并利用内外因式分解的优势。我们还考虑具有附加约束的同一问题，其中涉及f和g的某些变换，并确定这些约束是否强制解决方案具有唯一性。