Computation of Adaptive Fourier Series by Sparse Approximation of Exponential Sums

Journal of Fourier Analysis and Applications - In this paper, we study the convergence of adaptive Fourier sums for real-valued $$2\pi $$ -periodic functions. For this purpose, we approximate the...     

Application of the AAK theory and Prony-like Methods for sparse approximation of exponential sums

We derive a new method for optimal ℓ²-approximation of discrete signals on ℓ²(ℕ₀) 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) to derive an algorithm for computing a shorter exponential sum that approximates the original signal in the ℓ²-norm well. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)