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Application of the AAK theory for sparse approximation of exponential sums Pototskaia, Vlada


We derive a new method for optimal $\ell^2$-approximation of discrete signals on $\mathbb{N}_0$ whose entries can be represented as an exponential sum of finite length. Our approach employs Prony's method in a first step to recover the exponential sum that is determined by the signal. In the second step we use the theory of Adamjan, Arov and Krein (AAK-theory) to derive an algorithm for computing a shorter exponential sum that approximates the original signal in the $\ell^{2}$-norm well. AAK-theory originally determines best approximations of bounded periodic functions in Hardy-subspaces. We reformulate these ideas for our purposes and present the theory using only basic tools from linear algebra and Fourier analysis. The new algorithm is tested numerically in different examples. These results have been obtained jointly with Gerlind Plonka (Institute of Numerical and Applied Mathematics, University of G\"ottingen).

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