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 [51].
In the fundamental paper [52] the problem was formulated as follows. Let
${b}^{T}=\left[{b}_{1},{b}_{2},\dots ,{b}_{n}\right]$,
${c}^{T}=\left[{c}_{1},{c}_{2},\dots ,{c}_{n}\right]\in {\mathbb{\u2102}}^{n}$ or
(${\mathbb{\mathbb{R}}}^{n}$ 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

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 $S\subset \mathbb{\u2102}$, 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 ${\u2225\cdot \u2225}_{p}$–approximation
of a function $f\in {\text{L}}^{p}\left(0,\infty \right)$,
$1\le p\le \infty $ if