Vandermonde decomposition on intervals and its use for continuous compressed sensing

A classical result in mathematics dating back to the early twentieth century states that a rank-deficient, positive semidefinite, Toeplitz matrix admits a unique Vandermonde decomposition. This forms the basis of modern subspace and recent continuous compressed sensing (or gridless sparse) methods for frequency estimation. Conventionally, the Vandermonde decomposition is defined on the entire frequency domain; but in this paper, we show that the decomposition can be restricted to a prescribed frequency interval if the Toeplitz matrix also satisfies another linear matrix inequality. Besides solving an open classical moment problem in mathematics, this result is applied to practical frequency estimation scenarios in which the frequencies are known a priori to lie in certain frequency intervals. We show that the recent continuous compressed sensing methods derived based on the standard Vandermonde decomposition can be modified in a universal yet simple way to exploit this prior knowledge by applying the new Vandermonde decomposition result. Numerical results are provided to illustrate advantages of the proposed solutions.