Prony-Type Polynomials and Their Common Zeros - Prestin, Jürgen; Veselovska, Hanna (Frontiers in Applied Mathematics and Statistics, 2020)
The problem of hidden periodicity of a bivariate exponential sum $ {f({\bf n})=\sum_{j=1}^{N}a_j\exp{(-\mathrm{i}\langle{\bm \omega}_j, {\bf n}\rangle)}}, $ where $a_1,\dots,a_N \in \mathbb{C}\backslash\{0\}$ and ${\bf n}\in \mathbb{Z}^2$, is to recover frequency vectors ${\bm \omega}_1, \dots, {\bm \omega}_N \in [0,2\pi)^2$ using finitely many {samples of $f$.} Recently, this problem has received a lot of attention, and different approaches have been proposed to obtain its solution. For example, \cite{KunisProny} relies on the kernel basis analysis of the multilevel Toeplitz matrix of moments of $f$. In \cite{CuytProny2}, the exponential analysis has been considered as a Pad\'e approximation problem. In contrast to the previous method, the algorithms developed in \cite{BDAR,Cuyt} use sampling of $f$ along several lines in the hyperplane to obtain the univariate analog of the problem, which can be solved by classical one-dimensional approaches. Nevertheless, the stability of numerical solutions in the case of noise corruption still has a lot of open questions, especially when the number of parameters increases. Inspired by the one-dimensional approach developed in \cite{FMPrestin}, we propose to use the method of Prony-type polynomials, where the elements ${\bm \omega}_1, \dots, {\bm \omega}_N $ can be recovered due to a set of common zeros of the monic bivariate polynomial of an appropriate multi-degree. The use of Cantor pairing functions allows us to express bivariate Prony-type polynomials in terms of determinants and to find their exact algebraic representation. With respect to the number of samples the method of Prony-type polynomials is situated between the methods proposed in \cite{KunisProny, Cuyt}. Although the method of Prony-type polynomials requires more samples than \cite{Cuyt}, numerical computations show that the algorithm behaves more stable with regard to noisy data. Besides, combining the method of Prony-type polynomials with an autocorrelation sequence allows the improvement of the stability of the method in general.