]> 4.4 Discussion of results

### 4.4 Discussion of results

The existence and uniqueness problems of the best ${∥\cdot ∥}_{2}$–approximation have been analysed. An important question is the error estimation as a function of $n$. In the approximation theory such estimates are known as the Jackson–type inequalities. Since the maximum of $J$ at ${S}^{n}$ is greater than the maximum of $J$ at "the diagonal" ${\lambda }_{1}={\lambda }_{2}=\dots ={\lambda }_{n}$, then one of possible ways to get a Jackson–type inequality is to estimate the approximation error with the aid of the Laguerre functions, which follows from Remark 4.2.1. Such an idea was suggested also in [27, Section 6]. A derivation of a Jackson–type inequality requires considering the countable sequence ${\left\{{\lambda }_{i}\right\}}_{i=1}^{\infty }$ and thus it is naturally close to the question whether the system of exponentials ${\left\{{e}^{{\lambda }_{i}t}\right\}}_{i=1}^{\infty }$ forms Riesz basis in ${\text{L}}^{2}\left(0,\infty \right)$ , , .

Starting from the semigroups of shifts (or even weighted shifts) generators in other spaces, related to ${\text{L}}^{2}\left(0,\infty \right)$, one can get generalizations of the above results. For instance, in the space ${\ell }^{2}\left({\mathbb{ℤ}}^{\ast }\right)$, ${\mathbb{ℤ}}^{\ast }=\left\{0,1,2,\dots \right\}$ the semigroup of left–shifts generator has the form

 $\left({A}^{\ast }x\right)\left(j\right)=x\left(j+1\right),\phantom{\rule{2em}{0ex}}j=0,1,2,\dots$ (4.17)

This is a linear and bounded operator with the point spectrum ${\sigma }_{P}\left({A}^{\ast }\right)=D\subset \mathbb{ℂ}$, $D$ denotes the interior of a unit circle. The $k$–th normalized eigenfunction of ${A}^{\ast }$, corresponding to ${\lambda }_{k}\in {\sigma }_{P}\left({A}^{\ast }\right)$ is ${f}_{k}\left({\lambda }_{k}\right)\left(j\right)=\sqrt{1-{\left|{\lambda }_{k}\right|}^{2}}{\lambda }_{k}^{j}$, $j=0,1,2,\dots$. Here, we assume that ${\lambda }_{k}\ne 0$. If ${\lambda }_{k}=0$ then the vectors of the Cartesian basis in ${\ell }^{2}\left({\mathbb{ℤ}}^{\ast }\right)$ are generalized eigenvectors of ${A}^{\ast }$. Instead of the Hardy space ${\text{H}}^{2}\left({\Pi }^{+}\right)$ one should consider the Hardy space ${\text{H}}^{2}\left(\mathbb{𝔻}\right)$ , , . The formula (4.6) is still valid (see also [67, IX-4]) with the Malmquist functions being now of the form

 $\begin{array}{c}\hfill {ê}_{1}\left(\lambda ,z\right)=\frac{\sqrt{1-{\left|{\lambda }_{1}\right|}^{2}}}{1-{\lambda }_{1}z},\phantom{\rule{2em}{0ex}}{ê}_{k}\left(\lambda ,z\right)=\frac{\sqrt{1-{\left|{\lambda }_{k}\right|}^{2}}}{1-{\lambda }_{k}z}\phantom{\rule{0ex}{0ex}}\prod _{i=1}^{k-1}\frac{z-\overline{{\lambda }_{i}}}{1-{\lambda }_{i}z}\hfill \\ \hfill k=2,3,\dots ,n,\phantom{\rule{2em}{0ex}}\lambda =\left({\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{n}\right)\hfill \end{array}$ (4.18)

(these are the Taylor transforms of a system ${\left\{{e}_{k}\left(\lambda \right)\right\}}_{k=1}^{n}$ resulting from the system ${\left\{{f}_{k}\left({\lambda }_{k}\right)\right\}}_{k=1}^{n}$, by application of the Gram–Schmidt orthonormalization). The representation of the $k$–th Fourier coefficient looks as follows

 $\begin{array}{cc}\hfill 〈f,{e}_{k}\left(\lambda \right)〉=& \frac{1}{2\pi }{\int }_{-\pi }^{\pi }\stackrel{̂}{f}\left({e}^{j\theta }\right)\overline{{ê}_{k}\left(\lambda ,{e}^{j\theta }\right)}d\theta =\frac{1}{2\pi j}\underset{\left|z\right|=1}{\oint }\stackrel{̂}{f}\left(z\right)\overline{{ê}_{k}\left(\lambda ,\frac{1}{z}\right)}\frac{1}{z}dz=\hfill \\ \hfill =& \sum _{i=1}^{k}{Res}_{z=\overline{{\lambda }_{i}}}\left[\stackrel{̂}{f}\left(z\right)\frac{\sqrt{1-{\left|{\lambda }_{k}\right|}^{2}}}{z-\overline{{\lambda }_{k}}}\phantom{\rule{0ex}{0ex}}\prod _{i=1}^{k-1}\frac{1-{\lambda }_{i}z}{z-\overline{{\lambda }_{i}}}\right]\hfill \end{array}$ (4.19)

where $\stackrel{̂}{f}$, ${ê}_{k}$ denote the Taylor transforms of $f$, ${e}_{k}$, respectively. Erokhin , using the other representation for the Fourier coefficients, reduced formally the best ${\ell }^{2}\left({\mathbb{ℤ}}^{\ast }\right)$ – approximation problem with polynomials (equivalently the best ${\text{H}}^{2}\left(\mathbb{𝔻}\right)$ – approximation problem with rational functions) to the maximization of the function

$J:{\mathbb{𝔻}}^{n}\ni \lambda ↦\sum _{k=1}^{n}{\left|〈f,{e}_{k}\left(\lambda \right)〉\right|}^{2}.$

The proof technique worked out in Section 4.2, after some obvious modiﬁcation allow us to establish that the maximum of $J$ is attained not only on ${\mathbb{𝔻}}^{n}$ but also on ${S}^{n}$, where $S$ is a relatively closed subset of $\mathbb{𝔻}$ (as in Theorem 4.1.1).

The existence of a maximum of $J$ on ${\mathbb{𝔻}}^{n}$ has been proved also in ,  with the aid of yet another proof technique also taking advantage of the Hardy spaces, however without any explicit expressions for $J$.