]> 4.1 Introduction

### 4.1 Introduction

The best ${\text{L}}^{p}\left(0,\infty \right)$–approximation $\left(1\le p\le \infty \right)$ problem with exponential sums was the topic of numerous papers. An exhaustive bibliography of both theoretical foundations and applications is given in . In the fundamental paper  the problem was formulated as follows. Let ${b}^{T}=\left[{b}_{1},{b}_{2},\dots ,{b}_{n}\right]$, ${c}^{T}=\left[{c}_{1},{c}_{2},\dots ,{c}_{n}\right]\in {\mathbb{ℂ}}^{n}$ or (${\mathbb{ℝ}}^{n}$ in the case of approximation of a real function from the considered class). The solution ${Y}_{n}\left(b,c,\cdot \right)$ of the Cauchy problem
 ${y}^{\left(n\right)}\left(t\right)+{c}_{1}{y}^{\left(n-1\right)}\left(t\right)+\dots +{c}_{n}y\left(t\right)=0,\phantom{\rule{2em}{0ex}}{y}^{\left(j-1\right)}\left(0\right)={b}_{j},\phantom{\rule{2em}{0ex}}j=1,2,\dots ,n$ (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 ${Y}_{n}\left(b,c,\cdot \right)$. For any given set $S\subset \mathbb{ℂ}$, we consider ${V}_{n}\left(S\right)$ – the set of all exponential sums of order at most $n$, whose spectra are contained in $S$. An element ${y}_{0}\in {V}_{n}\left(S\right)$ is said to be the best ${∥\cdot ∥}_{p}$approximation of a function $f\in {\text{L}}^{p}\left(0,\infty \right)$, $1\le p\le \infty$ if

 ${∥f-{y}_{0}∥}_{p}={inf}_{y\in {V}_{n}\left(S\right)}{∥f-y∥}_{p}$ (4.2)

where

${∥g∥}_{p}=\left\{\begin{array}{cc}{\left[{\int }_{0}^{\infty }{\left|g\left(t\right)\right|}^{p}dt\right]}^{1∕p},\hfill & \hfill 1\le p\le \infty \\ {ess\phantom{\rule{3.26212pt}{0ex}}}_{}\hfill \end{array}\right\}$