Nonlinear approximation by sums of nonincreasing exponentials

Abstract

Many applications in electrical engineering, signal processing and mathematical physics lead to the following approximation problem: let h be a short linear combination of nonincreasing exponentials with complex exponents. Determine all exponents, all coefficients and the number of summands from finitely many equispaced sampled data of h. This is a nonlinear inverse problem. This article is an extension of Potts and Tasche [Parameter estimation for exponential sums by approximate Prony method, Signal Process., 90 (2010), pp. 1631–1642], where only the case of sums of complex exponentials with real frequencies is considered. In the following, we present new results on an approximate Prony method (APM) which is based on Beylkin and Monzón [On approximations of functions by exponential sums, Appl. Comput. Harmon. Anal. 19 (2005), pp. 17–48]. In contrast to their results, we apply perturbation theory for a singular value decomposition of a rectangular Hankel matrix such that we can describe the properties and the numerical behaviour of APM in detail. For this inverse problem, the number of sampled data acts as regularization parameter. The first part of APM recovers the exponents. The second part computes the coefficients by an overdetermined linear Vandermonde-type system. Numerical experiments show the performance of our method.

Keywords:

• nonlinear approximation
• exponential sum
• approximate Prony method
• singular value decomposition
• matrix perturbation theory
• perturbed rectangular Hankel matrix
• Vandermonde-type matrix

AMS Subject Classifications::

• 41A30
• 15A18
• 65F15
• 65F20
• 94A12

Acknowledgement

The first author gratefully acknowledges the support by the German Research Foundation within the project KU 2557/1-1.