the regularizing summands $\tau_m$ (if the initial system is regular, then it is natural to set $\sigma=\tau_m\equiv 0$). The task is now to find the numbers $\tau_m$ such that the conditions of Theorem \[th1\] are satisfied. Assume that we have done this, then by Theorem \[th1\] we obtain the interpolation identity $${\tilde f}(z)=H_{n}(\{\mu_k\},\{\lambda_k\},h;z)+O(z^{2n}).$$ Returning to the initial interpolation problem and taking into account (\[F(z)\]) yield $$\label{f_reg} f(z)=H_n(\{\mu_k\},\{\lambda_k\},h;z)-\sigma
H_{\nu}(\{\alpha_k\},\{r\beta_k\},h;z) +O(z^{2n}).$$ At the same time it is reasonable to choose ${\nu}$ as small as possible. However, there is a natural restriction on $\nu$, which is seen from the following statement (cf. §4 in [@Lyubich]).

\[ranks\] It is necessary for the regularity of the varied system $(\ref{Var_disc_mom_syst})$ that $${\nu}{\geqslant}n-{\operatorname{rank}}\left(s_{i+j-2}\right)_{i,j=1}^n.$$

Consider the Hankel matrices $$\mathbf{H}:=\left(s_{i+j-2}\right)_{i,j=1}^n, \qquad
\mathbf{R}:=\sigma\left(\sum_{k=1}^{\nu} \alpha_k (r
\beta_k)^{i+j-2}\right)_{i,j=1}^n.$$ For the regularity of the system $(\ref{Var_disc_mom_syst})$ it is necessary that the coefficient before $\lambda^n$ of the corresponding generating polynomial $G_n$ does not vanish, i.e. $\det (\mathbf{H}+\mathbf{R})\ne 0$, ${\operatorname{rank}}(\mathbf{H}+\mathbf{R})= n$. It is well known [@Horn §0.4.5] that for any $n\times n$-matrices $A$ and $B$ $$\label{rank_inequality} {\operatorname{rank}}(A+B){\leqslant}\min\{{\operatorname{rank}}A + {\operatorname{rank}}B;n\}.$$ Consequently, if the system $(\ref{Var_disc_mom_syst})$ is regular, then necessarily $n {\leqslant}{\operatorname{rank}}\mathbf{H} + {\operatorname{rank}}\mathbf{R}$. It remains to note that ${\operatorname{rank}}\mathbf{R}{\leqslant}{\nu}$. Indeed, the following representation is valid: $$\mathbf{R}=\sigma\sum_{k=1}^{\nu} 