Spectral Criteria for Unique Signal Recovery from Two-Sided Sampling

The identification of sampling sets that enable unique signal recovery is fundamental to many applications in signal processing and remains a central problem in mathematical analysis. Recent studies in the mathematical literature, particularly in the context of the Fourier transform and crystalline measures, have developed a theory that empowers signal recovery from two-sided sampling in both time and frequency domains. Kulikov, Nazarov, and Sodin introduced a method for identifying pairs of sets that enable unique recovery, based on functional inequalities of the Wirtinger-Poincar\'{e} type. In this work, we propose an alternative, spectral approach based on analogies with quantum mechanics. By relating uniqueness pairs to eigenvalue estimates of associated self-adjoint operators, our method offers a conceptually simpler and more flexible framework for studying signal recovery from two-sided sampling. Our approach extends naturally to other unitary transforms commonly used in signal processing. We demonstrate its effectiveness in the contexts of the Fractional Fourier transform and the Hankel transform.