]>
(4.1) |
is called an exponential sum of order if is not a solution of any problem (4.1) of order less than . The set of all roots of the characteristic polynomial of the system (4.1) is called the spectrum of the exponential sum . For any given set , we consider – the set of all exponential sums of order at most , whose spectra are contained in . An element is said to be the best –approximation of a function , if
(4.2) |
where
(the standard norm in ). Let , and .
Theorem 4.1.1. [52, Theorem 3, p. 392]. Any function , has the best -approximation in iff is closed in , if , and is closed in , if .
The proof is nonelementary and it does not provide an explicit construction of the best –approximation. To determine the best sum of exponentials one can take advantage of the necessary condition for (4.2) to hold [52, Theorem 4, p. 399]. In this way, practically verifiable analytic conditions characterizing the best approximate can be obtained in the case , . They are the essence of the next theorem.
Theorem 4.1.2. [52, Corollary 3, p. 402]. Let and be the best or locally best approximation of , . Then the generalized Aigrain–Williams equations are satisfied,
(4.3) |
where , denote the Laplace transforms of and , respectively, , , are such that
is the characteristic polynomial of the related system (4.1).
Aigrain and Williams initiated the extensive studies of the system (4.3) in the case , (see references in [51] and [52]). Let us mention that they were only motivated by simplification of formulae. The above assumptions seldom hold in practice.
The aim of this chapter is to derive explicit formulae for the – approximation error, from which a short proof of Theorem 4.1.1, as well as an evaluation method, alternative with respect to solving (4.3), will follow [33].