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(n)=_j=1^Na_j(-iω_j, n), where a_1,…,a_N C{0} and nZ^2, is to recover frequency vectors ω_1, …, ω_N [0,2π)^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, KunisProny relies on the kernel basis analysis of the multilevel Toeplitz matrix of moments of f. In CuytProny2, the exponential analysis has been considered as a Padé approximation problem. In contrast to the previous method, the algorithms developed in 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 FMPrestin, we propose to use the method of Prony-type polynomials, where the elements ω_1, …, ω_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 KunisProny, Cuyt. Although the method of Prony-type polynomials requires more samples than 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.