Hankel and Toeplitz operators of finite rank and Prony's problem in several variables

TL;DR

This paper explores the deep connections between finite-rank Hankel and Toeplitz operators and Prony's problem in multiple variables, revealing their equivalence and implications for polynomial ideals and signal processing.

Contribution

It establishes the fundamental equivalence between finite-rank Hankel operators, polynomial ideals, and Prony's problem in multiple variables, linking analytic, algebraic, and numeric approaches.

Findings

Finite-rank Hankel operators are essentially equivalent to solutions of Prony's problem.

Connections between moment problems, shift-invariant spaces, and annihilating ideals are elucidated.

Vandermonde matrix factorizations play a key role in understanding these operators.

Abstract

Prony's problem in several variables has attracted some attention recently and provides an interesting combination of polynomial ideal theory with analytic and numeric computations. This note points out further connections to Hankel operators of finite rank as they appear in multidimensional moment problems, shift invariance signal spaces, annihilating ideals of filters and factorization of the Hankel matrices and operators by means of Vandermonde matrices. In fact, it turns out that these concepts are essentially equivalent.