The classical result of Vandermonde decomposition of positive semidefinite Toeplitz matrices can date back to the early twentieth century. It forms the basis of modern subspace and recent atomic norm methods for frequency estimation. In this paper, we study the Vandermonde decomposition in which the frequencies are restricted to lie in a given interval, referred to as frequency-selective Vandermonde decomposition. The existence and uniqueness of the decomposition are studied under explicit conditions on the Toeplitz matrix. The new result is connected by duality to the positive real lemma for trigonometric polynomials nonnegative on the same frequency interval. Its applications in the theory of moments and line spectral estimation are illustrated. In particular, it provides a solution to the truncated trigonometric K-moment problem. It is used to derive a primal semidefinite program formulation of the frequency-selective atomic norm in which the frequencies are known a priori to lie in a certain frequency band. Numerical examples are also provided.
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.