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BIRS Workshop Lecture Videos

Polynomial-exponential decomposition from moments Mourrain, Bernard


Several problems with applications in signal processing, functional approximation involve series representations, which are sums of polynomial-exponential functions. Their decomposition often allows to recover the structure of the underlying signal or data. Using the duality between polynomials and formal power series, we will describe the correspondence between polynomial-exponential series and Artinian Gorenstein algebras. We will give a generalisation of Kronecker theorem to the multivariate case, showing that the symbol of a Hankel operator of finite rank is a polynomial-exponential series and by connecting the rank of the Hankel operator with the decomposition of the symbol. We will describe algorithms for computing the polynomial-exponential decomposition of a series from its first coefficients, exploiting eigenvector methods for solving polynomial equations and a Gram-Schmidt orthogonalization process. A key ingredient of the approach is the flat extension criteria, which leads to a rank condition for a Carath{\'e}odory-Fej{\'e}r decomposition of multivariate Hankel matrices. The approach will be illustrated in different contexts: sparse interpolation of polylog functions from values, decomposition of polynomial-exponential functions from values, tensor decomposition, decomposition of symbols of convolution (or cross-correlation) operators of finite rank, reconstruction of measures as weighted sums of Dirac measures from moments. Numerical computation will be given to show the behaviour of the method.

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