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This paper studies the continuous-domain inverse problem of recovering Radon measures on the one-dimensional torus from low-frequency Fourier coefficients, where K c is the cutoff frequency. Our approach consists in minimizing the total-variation norm among all Radon measures that are consistent with the observations. We call this problem the basis pursuit in the continuum (BPC). We characterize the solution set of (BPC) in terms of uniqueness and describe its sparse solutions which are sums of few signed Dirac masses. The characterization is determined by the spectrum of a Toeplitz and Hermitian-symmetric matrix that solely depends on the observations. More precisely, we prove that (BPC) has a unique solution if and only if this matrix is neither positive definite nor negative definite. If it has both a positive and negative eigenvalue, then the unique solution is the sum of at most 2K c Dirac masses, with at …