]> 5.1 Introduction

5.1 Introduction

In the previous section we have considered the problem of approximation of a given function f L2(0,) with exponential sums, i.e., the problem of finding g Mn such that
f gL2(0,) = min yMn f yL2(0,) ,

where Mn is the n–dimensional subspace of L2(0,) spanned by an n–tuple of normalized eigenvectors of the operator

𝒜h = h,D(𝒜) = W1,2[0,)

corresponding to eigenvalues {λk}k=1n Π. By the orthogonal projection theorem [86, Theorem 3.6, p. 38], the best approximant of f in Mn is the orthogonal projection of f on Mn. To be more precise,

min gMn f gL2(0,) = f PnfL2(0,)

where Pn stands for the orthoprojector onto Mn. The orthoprojector Pn can be expressed in terms of the Malmquist functions and is given by (4.7) and (4.9).

In this section we give an alternative approach, more convenient for application of the standard mathematical software. The problem of convergence as n is discussed from both theoretical and numerical viewpoint. The results are illustrated by the problem of finding the optimal adjustment of the proportional controller stabilizing a distributed plant [39].