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Atomic norm minimization for decomposition into complex exponentials and optimal transport in Fourier domain. (English) Zbl 1455.42002
In this paper, atomic decompositions of complex vectors \(v = (v_m)_{m=-M}^M\) are studied. An atom is a vector \(a(z) = \big(\exp(- {\mathrm i}\,m\,{\mathrm{arg}}\,z)\big)_{m=-M}^M\) with \(z \in {\mathbb T} = \{z \in \mathbb C : \,|z|=1\}\). If the vector \(v\) has the atomic decomposition \[ v = \sum_{k=1}^K c_k \,a(z_k) \] with distinct \(z_k \in \mathbb T\) and \(c_k \in {\mathbb R} \setminus \{0\}\), then the discrete measure \[ \mu = \sum_{k=1}^K c_k \,\delta(z_k) \] explains \(v\), where \(\delta(z_k)\) is the Dirac measure. The author reviews properties of atomic decompositions, the close relationship with positive semidefinite Toeplitz matrices, and the Prony method to extract the parameters from a linear combination of atoms. He presents a characterization of the existence and uniqueness of a measure explaining \(v\) with minimal total variation norm. This minimal measure is numerically determined by a convex semidefinite program and Prony’s method. Further, the author explores the optimal transport between two measures, of which only a finite sequence of Fourier coefficients is known.
42A10 Trigonometric approximation
42A70 Trigonometric moment problems in one variable harmonic analysis
65K05 Numerical mathematical programming methods
65T50 Numerical methods for discrete and fast Fourier transforms
90C22 Semidefinite programming
90C25 Convex programming
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