Spectral Criteria for Unique Signal Recovery from Two-Sided Sampling
Oleg Szehr
TL;DR
The paper develops a spectral framework for uniqueness of signal recovery from two-sided sampling by linking uniqueness pairs to eigenvalue bounds of confined, positive self-adjoint operators. It recasts Wirtinger–Poincaré-based arguments into a Rayleigh-quotient approach, and extends the theory from the classical Fourier transform to a broad class of unitary transforms, notably the Fractional Fourier and Hankel transforms. Through a general operator $X=\tfrac{1}{2}(M_{x^2}+U^*M_{x^2}U)$ and its confined analogues, the work derives conditions—via ground-state energy bounds and density-like inequalities—that guarantee that dual-domain samples force the function to vanish, while also clarifying when these conditions fail. The results yield a coherent, extensible toolkit for two-sided sampling problems in time–frequency analysis with explicit criteria that interpolate between minimal and dense sampling regimes as transform parameters vary. This framework broadens the applicability of unique-recovery results and provides concrete spectral methods for analyzing unitary transforms in signal processing.
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 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-Poincaré 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.