]> 5.1 Introduction

### 5.1 Introduction

In the previous section we have considered the problem of approximation of a given function $f\in {\text{L}}^{2}\left(0,\infty \right)$ with exponential sums, i.e., the problem of ﬁnding $g\in {M}_{n}$ such that
${∥f-g∥}_{{\text{L}}^{2}\left(0,\infty \right)}={min}_{y\in {M}_{n}}{∥f-y∥}_{{\text{L}}^{2}\left(0,\infty \right)},$

where ${M}_{n}$ is the $n$–dimensional subspace of ${\text{L}}^{2}\left(0,\infty \right)$ spanned by an $n$–tuple of normalized eigenvectors of the operator

${\mathcal{𝒜}}^{\ast }h={h}^{\prime },\phantom{\rule{2em}{0ex}}D\left({\mathcal{𝒜}}^{\ast }\right)={\text{W}}^{1,2}\left[0,\infty \right)$

corresponding to eigenvalues ${\left\{{\lambda }_{k}\right\}}_{k=1}^{n}\subset {\Pi }^{-}$. By the orthogonal projection theorem [86, Theorem 3.6, p. 38], the best approximant of $f$ in ${M}_{n}$ is the orthogonal projection of $f$ on ${M}_{n}$. To be more precise,

${min}_{g\in {M}_{n}}{∥f-g∥}_{{\text{L}}^{2}\left(0,\infty \right)}={∥f-{P}_{n}f∥}_{{\text{L}}^{2}\left(0,\infty \right)}$

where ${P}_{n}$ stands for the orthoprojector onto ${M}_{n}$. The orthoprojector ${P}_{n}$ 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 $n\to \infty$ 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 .