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 ${\ell}^{2}\left({\mathbb{\mathbb{Z}}}^{\ast}\right)$, ${\mathbb{\mathbb{Z}}}^{\ast}=\left\{0,1,2,\dots \right\}$ 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 ${\sigma}_{P}\left({A}^{\ast}\right)=D\subset \mathbb{\u2102}$, $D$ 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 ${\ell}^{2}\left({\mathbb{\mathbb{Z}}}^{\ast}\right)$ are generalized eigenvectors of ${A}^{\ast}$. Instead of the Hardy space ${\text{H}}^{2}\left({\Pi}^{+}\right)$ one should consider the Hardy space ${\text{H}}^{2}\left(\mathbb{\mathbb{D}}\right)$ [25], [49], [67]. The formula (4.6) is still valid (see also [67, IX-4]) with the Malmquist functions being now of the form

(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

where $\widehat{f}$, ${\xea}_{k}$ denote the Taylor transforms of $f$, ${e}_{k}$, respectively. Erokhin [24], using the other representation for the Fourier coefficients, reduced formally the best ${\ell}^{2}\left({\mathbb{\mathbb{Z}}}^{\ast}\right)$ – approximation problem with polynomials (equivalently the best ${\text{H}}^{2}\left(\mathbb{\mathbb{D}}\right)$ – approximation problem with rational functions) to the maximization of the function

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

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 ${\mathbb{\mathbb{D}}}^{n}$ but also on ${S}^{n}$, where $S$ is a relatively closed subset of $\mathbb{\mathbb{D}}$ (as in Theorem 4.1.1).

The existence of a maximum of $J$ on ${\mathbb{\mathbb{D}}}^{n}$ has been proved also in [74], [2] with the aid of yet another proof technique also taking advantage of the Hardy spaces, however without any explicit expressions for $J$.