We derive bounds on the extremal singular values and the condition number of N × K, with N K, Vandermonde matrices with nodes in the unit disk. The mathematical techniques we develop to prove our main results are inspired by a link-first established by Selberg [1] and later extended by Moitra [2]-between the extremal singular values of Vandermonde matrices with nodes on the unit circle and large sieve inequalities. Our main conceptual contribution lies in establishing a connection between the extremal singular values of Vandermonde matrices with nodes in the unit disk and a novel large sieve inequality involving polynomials in z ∈ C with |z| 1. Compared to Bazán's upper bound on the condition number [3], which, to the best of our knowledge, constitutes the only analytical result-available in the literature-on the condition number of Vandermonde matrices with nodes in the unit disk, our bound not only takes a much simpler form, but is also sharper for certain node configurations. Moreover, the bound we obtain can be evaluated consistently in a numerically stable fashion, whereas the evaluation of Bazán's bound requires the solution of a linear system of equations which has the same condition number as the Vandermonde matrix under consideration and can therefore lead to numerical instability in practice. As a byproduct, our result-when particularized to the case of nodes on the unit circle-slightly improves upon the Selberg-Moitra bound.

While spike trains are obviously not band-limited, the theory of super-resolution tells us that perfect recovery of unknown spike locations and weights from low-pass Fourier transform measurements is possible provided that the minimum spacing, ∆, between spikes is not too small. Specifically, for a measurement cutoff frequency of f c , Donoho [2] showed that exact recovery is possible if the spikes (on R) lie on a lattice and ∆ > 1/f c , but does not specify a corresponding recovery method. Candès and Fernandez-Granda [3, 4] provide a convex programming method for the recovery of periodic spike trains (i.e., spike trains on the torus T), which succeeds provably if ∆ > 2/f c and f c 128 or if ∆ > 1.26/f c and f c 10 3 , and does not need the spikes within the fundamental period to lie on a lattice. In this paper, we develop a theory of super-resolution from short-time Fourier transform (STFT) measurements. Specifically, we present a recovery method similar in spirit to the one in [3] for pure Fourier measurements. For a STFT Gaussian window function of width σ = 1/(4f c ) this method succeeds provably if ∆ > 1/f c , without restrictions on f c . Our theory is based on a measure-theoretic formulation of the recovery problem, which leads to considerable generality in the sense of the results being grid-free and applying to spike trains on both R and T. The case of spike trains on R comes with significant technical challenges. For recovery of spike trains on T we prove that the correct solution can be approximated-in weak-* topology-by solving a sequence of finite-dimensional convex programming problems.

Performance analyses of subspace algorithms for cisoid parameter estimation available in the literature are predominantly of statistical nature with a focus on asymptoticeither in the sample size or the SNR-statements. This paper presents a deterministic, finite sample size, and finite-SNR performance analysis of the ESPRIT algorithm and the matrix pencil method. Our results are based, inter alia, on a new upper bound on the condition number of Vandermonde matrices with nodes inside the unit disk. This bound is obtained through a generalization of Hilbert's inequality frequently used in large sieve theory.

While spike trains are obviously not band-limited, the theory of super-resolution tells us that perfect recovery of unknown spike locations and weights from low-pass Fourier transform measurements is possible provided that the minimum spacing, Δ, between spikes is not too small. Specifically, for a cutoff frequency of f c , Donoho [2] shows that exact recovery is possible if Δ > 1/f c , but does not specify a corresponding recovery method. On the other hand, Candès and Fernandez-Granda [3] provide a recovery method based on convex optimization, which provably succeeds as long as Δ > 2/f c . In practical applications one often has access to windowed Fourier transform measurements, i.e., short-time Fourier transform (STFT) measurements, only. In this paper, we develop a theory of super-resolution from STFT measurements, and we propose a method that provably succeeds in recovering spike trains from STFT measurements provided that Δ > 1/f c .

Abstract-We formulate a unified framework for the separation of signals that are sparse in "morphologically" different redundant dictionaries. This formulation incorporates the socalled "analysis" and "synthesis" approaches as special cases and contains novel hybrid setups. We find corresponding coherencebased recovery guarantees for an 1-norm based separation algorithm. Our results recover those reported in Studer and Baraniuk, ACHA, submitted, for the synthesis setting, provide new recovery guarantees for the analysis setting, and form a basis for comparing performance in the analysis and synthesis settings. As an aside our findings complement the D-RIP recovery results reported in Candès et al., ACHA, 2011, for the "analysis" signal recovery problem minimize x Ψ x 1 subject to y − A x 2 ≤ ε by delivering corresponding coherence-based recovery results.