]> 4.1 Introduction

4.1 Introduction

The best Lp(0,)–approximation (1 p ) problem with exponential sums was the topic of numerous papers. An exhaustive bibliography of both theoretical foundations and applications is given in [51]. In the fundamental paper [52] the problem was formulated as follows. Let bT = [b 1,b2,,bn], cT = [c 1,c2,,cn] n or (n in the case of approximation of a real function from the considered class). The solution Y n(b,c,) of the Cauchy problem
y(n)(t) + c 1y(n1)(t) + + c ny(t) = 0,y(j1)(0) = b j,j = 1, 2,,n (4.1)

is called an exponential sum of order n if Y n is not a solution of any problem (4.1) of order less than n. The set of all roots of the characteristic polynomial of the system (4.1) is called the spectrum of the exponential sum Y n(b,c,). For any given set S , we consider V n(S) – the set of all exponential sums of order at most n, whose spectra are contained in S. An element y0 V n(S) is said to be the best papproximation of a function f Lp(0,), 1 p if

f y0p = inf yV n(S) f yp (4.2)

where

gp = 0g(t) pdt1p,1 p esssup t0 g(t) , p =

(the standard norm in Lp). Let Π := {s : Re s < 0}, and Π+ := {s : Re s > 0}.

Theorem 4.1.1. [52, Theorem 3, p. 392]. Any function f Lp(0,), 1 p has the best p-approximation in V n(S) iff S Π is closed in Π, if 1 p < , and S Π¯ is closed in Π¯, if p = .

The proof is nonelementary and it does not provide an explicit construction of the best p–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 p = 2, S = Π¯. They are the essence of the next theorem.

Theorem 4.1.2. [52, Corollary 3, p. 402]. Let f L2(0,) and y0 be the best or locally best approximation of f V n(Π), f y0p > 0. Then the generalized Aigrain–Williams equations are satisfied,

ŷ0(i1)(λ¯ j) = f̂(i1)(λ¯ j),i = 1, 2,, 2kj,j = 1, 2,,l (4.3)

where ŷ0, f̂ denote the Laplace transforms of y0 and f, respectively, l, kj, λj are such that

(z λ1)k1 (z λ2)k2 (z λl)kl ,k1 + k2 + + kl = n

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 l = n, λ1λ2λn (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 2 – 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].