Motivated by the classical Pauli problem of determining a function $f$ (up to global phase) from the magnitudes $|f|,|\widehat{f}|$, we shall investigate a related question: given the values of $|f|$ and $|\widehat{f}|$ on discrete sequences, when is it possible to recover the values of $|f|,|\widehat{f}|$ on the whole real line?

By employing techniques recently developed in the context of Fourier Uniqueness Pairs, we shall see several necessary and sufficient conditions for the property above to hold. Time-permitting, we shall also see how such results generalize to higher dimensions and have direct consequences to problems of recovery-type for partial differential equations.

This talk is based on joint work with Mateus Sousa.