Matrix analysis
Fast Fourier transforms for nonequispaced data
SIAM Journal on Scientific Computing
Conditioning of Rectangular Vandermonde Matrices with Nodes in the Unit Disk
SIAM Journal on Matrix Analysis and Applications
A Stable Numerical Method for Inverting Shape from Moments
SIAM Journal on Scientific Computing
Fast communication: A Prony-like method for non-uniform sampling
Signal Processing
Stability Results for Scattered Data Interpolation by Trigonometric Polynomials
SIAM Journal on Scientific Computing
Using NFFT 3---A Software Library for Various Nonequispaced Fast Fourier Transforms
ACM Transactions on Mathematical Software (TOMS)
Parameter estimation for exponential sums by approximate Prony method
Signal Processing
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In this paper, we discuss the numerical solution of two nonlinear approximation problems. Many applications in electrical engineering, signal processing, and mathematical physics lead to the following problem. Let $h$ be a linear combination of exponentials with real frequencies. Determine all frequencies, all coefficients, and the number of summands if finitely many perturbed, uniformly sampled data of $h$ are given. We solve this problem by an approximate Prony method (APM) and prove the stability of the solution in the square and uniform norm. Further, an APM for nonuniformly sampled data is proposed too. The second approximation problem is related to the first one and reads as follows: Let $\varphi$ be a given 1-periodic window function as defined in section 4. Further let $f$ be a linear combination of translates of $\varphi$. Determine all shift parameters, all coefficients, and the number of translates if finitely many perturbed, uniformly sampled data of $f$ are given. Using Fourier technique, this problem is transferred into the above parameter estimation problem for an exponential sum which is solved by APM. The stability of the solution is discussed in the square and uniform norm too. Numerical experiments show the performance of our approximation methods.