We derive a new generalization of Prony's method to reconstruct M-sparse expansions of (generalized) eigenfunctions of linear operators from only O(M) suitable values in a deterministic way. The proposed method covers the wellknown reconstruction methods for M-sparse sums of exponentials as well as for the interpolation of M-sparse polynomials by using special linear operators in C(R). Further, we can derive new reconstruction formulas for M-sparse expansions of orthogonal polynomials using the Sturm-Liouville operator. The method is also applied to the recovery of M-sparse vectors in finite-dimensional vector spaces.

Prony's method is a prototypical eigenvalue analysis based method for the reconstruction of a finitely supported complex measure on the unit circle from its moments up to a certain degree. In this note, we give a generalization of this method to the multivariate case and prove simple conditions under which the problem admits a unique solution. Provided the order of the moments is bounded from below by the number of points on which the measure is supported as well as by a small constant divided by the separation distance of these points, stable reconstruction is guaranteed. In its simplest form, the reconstruction method consists of setting up a certain multilevel Toeplitz matrix of the moments, compute a basis of its kernel, and compute by some method of choice the set of common roots of the multivariate polynomials whose coefficients are given in the second step. All theoretical results are illustrated by numerical experiments.

Abstract. 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 ϕ be a given 1-periodic window function as defined in section 4. Further let f be a linear combination of translates of ϕ. 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.

In this work we present two sparse deconvolution methods for nondestructive testing. The first method is a special matching pursuit (MP) algorithm in order to deconvolve the mixed data (signal and noise), and thus to remove the unwanted noise. The second method is based on the approximate Prony method (APM). Both methods employ the sparsity assumption about the measured ultrasonic signal as prior knowledge. The MP algorithm is used to derive a sparse representation of the measured data by a deconvolution and subtraction scheme. An orthogonal variant of the algorithm (OMP) is presented as well. The APM technique also relies on the assumption that the desired signals are sparse linear combinations of (reflections of) the transmitted pulse. For blind deconvolution, where the transducer impulse response is unknown, we offer a general Gaussian echo model whose parameters can be iteratively adjusted to the real measurements. Several test results show that the methods work well even for high noise levels. Fur

The parameter reconstruction problem in a sum of Dirac measures from its low frequency trigonometric moments is well understood in the univariate case and has a sharp transition of identifiability with respect to the ratio of the separation distance of the parameters and the order of moments. Towards a similar statement in the multivariate case, we present an Ingham inequality which improves the previously best known dimension-dependent constant from square-root growth to a logarithmic one. Secondly, we refine an argument that an Ingham inequality implies identifiability in multivariate Prony methods to the case of commonly used max-degree by a short linear algebra argument, closely related to a flat extension principle and the stagnation of a generalized Hilbert function.

The matrix pencil method is an eigenvalue based approach for the parameter identification of sparse exponential sums. We derive a reconstruction algorithm for multivariate exponential sums that is based on simultaneous diagonalization. Randomization is used and quantified to reduce the simultaneous diagonalization to the eigendecomposition of a single random matrix. To verify feasibility, the algorithm is applied to synthetic and experimental fluorescence microscopy data.2010 Mathematics Subject Classification. 65T40, 42C15, 30E05, 65F30.

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The problem of recovering translates and corresponding amplitudes of sparse sums of Gaussians out of sampling values as well as reconstructing sparse sums of exponentials are nonlinear inverse problems that can be solved for example by Prony's method. Here, we want to demonstrate a new extension to multivariate input data.