Spectral Criteria for Uniqueness Pairs of Unitary Transforms

Abstract

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, particularly in the context of the Fourier transform and crystalline measures, have developed a theory of recovery from two-sided sampling, where samples are prescribed simultaneously in the physical and transformed domains. Kulikov, Nazarov, and Sodin introduced a method for identifying such uniqueness pairs based on functional inequalities of the Wirtinger-Poincar\'e type. In this work, we propose an alternative spectral approach motivated by quantum mechanics. The guiding observation is that zeros of a function and of its transform impose Dirichlet-type confinement in two conjugate representations, thereby converting two-sided uniqueness questions into lower-bound problems for confined Hamiltonians. For the Fourier transform, the relevant Hamiltonian is the harmonic oscillator, and the uniformly supercritical uniqueness criterion is recovered through a variational spectral argument. Our viewpoint extends to unitary transforms whose associated localization operators admit local Sturm-Liouville or Schr\"odinger-type confined realizations, a class that includes transforms commonly used in signal processing and mathematical physics. It abstracts the Wirtinger-Poincar\'e mechanism by replacing the ordinary Dirichlet-Laplacian constant with the local spectral floor of a Hamiltonian-type operator associated with the transform. We formulate this principle for Sturm-Liouville operators with weights or nontrivial coefficients, and illustrate it for the fractional Fourier transform and the Hankel transform, where phase-space rotation and singular endpoint behavior enter the uniqueness criteria.