Prony's method under an almost sharp multivariate Ingham inequality
For researchers in signal processing and inverse problems, this provides sharper theoretical guarantees for parameter recovery from low-frequency moments, though the improvement is incremental.
The authors improve the dimension-dependent constant in a multivariate Ingham inequality from square-root to logarithmic growth, and use this to refine identifiability guarantees for multivariate Prony methods under max-degree moment conditions.
The parameter reconstruction problem in a sum of Dirac measures from its low frequency trigonometric moments is well understood in the univariate case and has a sharp transition of identifiability with respect to the ratio of the separation distance of the parameters and the order of moments. Towards a similar statement in the multivariate case, we present an Ingham inequality which improves the previously best known dimension-dependent constant from square-root growth to a logarithmic one. Secondly, we refine an argument that an Ingham inequality implies identifiability in multivariate Prony methods to the case of commonly used max-degree by a short linear algebra argument, closely related to a flat extension principle and the stagnation of a generalized Hilbert function.