Injectivity of sampled Gabor phase retrieval in spaces with general integrability conditions

Abstract

It was recently shown that functions in $L^4([-B,B])$ can be uniquely recovered up to a global phase factor from the absolute values of their Gabor transforms sampled on a rectangular lattice. We prove that this remains true if one replaces $L^4([-B,B])$ by $L^p([-B,B])$ with $p\in [1,\infty ]$. To do so, we adapt the original proof by Grohs and Liehr and use a classical sampling result due to Beurling. Furthermore, we present a minor modification of a result of Müntz-Szász type by Zalik. Finally, we consider the implications of our results for more general function spaces obtained by applying the fractional Fourier transform to $L^p([-B,B])$ and for more general nonuniform sampling sets.

12 pages; major revision, generalised main result, shortened presentation

References in corpus (5)

• Uniqueness of STFT phase retrieval for bandlimited functions
• On foundational discretization barriers in STFT phase retrieval
• Injectivity of Gabor phase retrieval from lattice measurements
• Phase retrieval from sampled Gabor transform magnitudes: Counterexamples
• Sampling at twice the Nyquist rate in two frequency bins guarantees uniqueness in Gabor phase retrieval

Cited by in corpus (3)

• Phase retrieval of entire functions and its implications for Gabor phase retrieval
• Arithmetic progressions and holomorphic phase retrieval
• Translation-based completeness on compact intervals