Super-resolution via Prony-Type Polynomials
Anna Veselovska, Jürgen Prestin · 2025
The problem of hidden periodicity in three dimensions is to recover frequency vectors ω1, …, ωN∈ [0, 2π)3using finitely many samples of the exponential $f({\mathbf{n}}) = \sum olimits_{j = 1}^N {{a_j}} \exp \left( { - {\text{i}}\left\langle {{{\mathbf{\omega }}_j},{\mathbf{n}}} \right\rangle } \right)$, where a1, …, aN∈ ℂ\{0} and n ∈ ℤ3. Inspired by the approaches developed in [11], [30], we consider specifically constructed polynomials, which are called Prony-type polynomials, and show that the frequency vectors ω1, …, ωNcan be recovered via a set of common zeros of such polynomials. By employing Cantor tuple functions, we position the method of Prony-type polynomials within the spectrum of sampling requirements between the methods proposed in [21], [22]. While the Prony-type polynomial method demands more samples than the approach in [22], numerical experiments indicate that it exhibits greater stability in the presence of noisy data.