On approximation of functions by exponential sums

Parameter estimation for exponential sums by approximate Prony method

“…Using an analog of the Theorem of H. Weyl (see [8, Recently, a very interesting approach is described by G. Beylkin and L. Monzón [1], where the more general problem of approximation by exponential sums is considered. In contrast to [1], we consider a perturbed rectangular Hankel matrix (3.2).…”

“…Note that the solution of the overdetermined linear Vandermonde-type system is closely related to the inverse nonequispaced fast Fourier transform studied recently by the authors [4]. In contrast to [1], we present an approximate Prony method (APM) by means of matrix perturbation theory such that we can describe the properties and the numerical behavior of the APM in detail. The first part of APM estimates the frequencies and the second part solves an overdetermined linear Vandermonde-type system in a stable way.…”

“…In Section 2, we sketch the classical Prony method. Then in Section 3, we generalize a first APM based on ideas of G. Beylkin and L. Monzón [1]. The Sections 4 and 5 form the core of this paper with new results on APM.…”

“…[3] and the references therein. Our results are based on the paper [1] of G. Beylkin and L. Monzón. The nonlinear problem of finding the frequencies and coefficients can be split into two problems.…”

“…This is a nonlinear inverse problem. In this paper, we present new results on an approximate Prony method (APM) which is based on [1]. In contrast to [1], we apply matrix perturbation theory such that we can describe the properties and the numerical behavior of the APM in detail.…”