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Let $f, g \in L^2(\mathbb{R})$.

Is it true that if both $|f|=|g|$ and $|\hat f|=|\hat g|$ hold, then there exists $\theta \in \mathbb{R}$ such that $f=ge^{i\theta}$?

I am not able to prove it or disprove it. I suspect that this is true. Do you have a reference for this?

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This is true. This was proved by Hardy and Littlewood and the proof is reproduced in Zygmund's Trigonometric Series (which I don't have access to at the moment).

(Contradicting my prior "answer")

The answer to this question is negative. Such counterexamples are known as "Pauli partners" and are studied in, among other places, the quantum mechanics literature.

See, for example:

(J. V. Corbett and C. A. Hurst) Are Wave Functions Uniquely Determined by their position and momentum distributions? [https://www.cambridge.org/core/services/aop-cambridge-core/content/view/228E4A34D0B3C63C54B1A01006278C42/S0334270000001569a.pdf/are-wave-functions-uniquely-determined-by-their-position-and-momentum-distributions.pdf]

(P. Jaming) Phase Retrieval Techniques for Radar Ambiguity Problems [https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.521.4906&rep=rep1&type=pdf]

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  • $\begingroup$ Thank you. Do you know in which Chapter it is located (Trigonometric Series is about 600 pages long!) $\endgroup$
    – J.Mayol
    Commented Sep 1, 2021 at 6:43
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    $\begingroup$ I searched Zygmund's books but did not find anything. $\endgroup$
    – J.Mayol
    Commented Sep 2, 2021 at 9:40
  • $\begingroup$ Sorry I misread your question. I was thinking of Hardy and Littlewood's proof that characters are the only extremals to the Hausdorff-Young inequality on the circle, which includes an analysis of a question similar to yours. $\endgroup$
    – Mark Lewko
    Commented Sep 3, 2021 at 3:47
  • $\begingroup$ The answer turns out to be negative. I've adjusted my answer accordingly. $\endgroup$
    – Mark Lewko
    Commented Sep 7, 2021 at 15:52

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