Application of the AAK theory for sparse approximation of exponential\n sums

2016/09/30 by Gerlind Plonka, Plonka, Gerlind, Vlada Pototskaia +1
Mathematics · #15A18 #41A30 #42A16 #65F15 #Algebraic and Geometric Analysis #FOS: Mathematics #Mathematical Analysis and Transform Methods #Mathematical Approximation and Integration #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1609.09603

openalex publication_date 2016/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

In this paper, we derive a new method for optimal \ℓ¹- and\n\ℓ²-approximation of discrete signals on mathbb N₀ whose entries\ncan be represented as an exponential sum of finite length. Our approach employs\nProny's method in a first step to recover the exponential sum that is\ndetermined by the signal. In the second step we use the AAK-theory to derive an\nalgorithm for computing a shorter exponential sum that approximates the\noriginal signal in the \ℓ^p-norm well. AAK-theory originally determines\nbest approximations of bounded periodic functions in Hardy-subspaces. We\nrewrite these ideas for our purposes and give a proof of the used AAK theorem\nbased only on basic tools from linear algebra and Fourier analysis. The new\nalgorithm is tested numerically in different examples.\n

Citations

• On two problems concerning linear transformations in hilbert space
• On approximation of functions by exponential sums

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