On the Foundation of Prony's Method

Abstract

The paper concerns the foundation of Prony's method whose basis is the groundless assumption that the investigated function can be expanded into the sum of complex exponents. To begin with the class of continuous finite order functions is defined. The notion of the canonical expansion of a function into the formants is introduced. It is established that the formant's structure depends upon the type of the roots of the minimal annulating polynomials of the shift operators acting in the formant's shifts space. The formants over the field of complex numbers are obtained to be complex exponents in the case of simple roots and complex exponents multiplied by the polynomials in the case of multiple roots, what agrees with the assumption of Prony's method. The estimation of the spectral density of a stationary random process by Prony's method is considered. The presented foundation allows one not only to understand the method better, but to formulate its extensions.