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Nonlinear approximation by sums of nonincreasing exponentials. (English) Zbl 1222.41025
The authors study the inverse problem to determine the exponents, coefficients and number of summands of a linear combination of nonincreasing exponentials from finitely many equispaced sample data. The exact representation is the goal of classical Prony’s method which is notorious for its sensitivity to noise. The results of G. Beylkin and L. Monzon [Appl. Comput. Harmon. Anal. 19, No. 1, 17–48 (2005; Zbl 1075.65022] indicate that avoiding exact representations and incorporating an arbitrary but fixed accuracy help to control the ill-conditioning of the problem and to reduce the number of terms needed in the approximation. The authors present new results on an approximate Prony method and propose new algorithms. The properties and the numerical behavior of the method are analyzed in detail by applying perturbation theory for singular value decompositions of Hankel matrices. Numerical experiments show the performance of the method.

MSC:
41A30 Approximation by other special function classes
15A18 Eigenvalues, singular values, and eigenvectors
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65F20 Numerical solutions to overdetermined systems, pseudoinverses
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
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