$M<2n$) and the corresponding Padé interpolation, they do not arise in the present paper as we solve the interpolation problem of the order higher than $M$ with the same number of independent parameters $\{\mu_k\}_{k=1}^n$ and $\{\lambda_k\}_{k=1}^n$.

Nevertheless, the systems (\[SRS\_M\]) (consistent and even inconsistent) are of independent interest, moreover, they are actively studied in numerical analysis. Various methods for finding an approximate solution (in different senses) to them were developed, for example, in [@Beylkin; @Kung_residual; @Beylkin2; @Potts; @YF] (characteristics of one such method are discussed in Section \[Section6\]).

Amplitude and frequency operators in classical problems {#par2}
=======================================================

We now consider several classical problems, which are bound up with the class of amplitude and frequency operators.

[**.1. Hamburger moment problem.**]{} Theorem \[th1\] raises the question about interpolating amplitude and frequency operators with real $\lambda_k$ and $\mu_k$ (in particular, with $\mu_k>0$). This question is well-studied [@Akhiezer1 Ch. 2] and can be settled by discretization of the following classical Hamburger problem: given a sequence of real numbers $\{s_m\}$, $m\in
\mathbb{N}_0$, find a non-negative Borel measure $\mu$ on $\mathbb{R}$ such that $$\label{Hamburger} s_m = \int_{-\infty}^\infty \lambda^m\,d
\mu(\lambda),\qquad m\in \mathbb{N}_0.$$ Namely, the following criterion is valid [@Akhiezer1 Ch. 2, §1]: the problem (\[Hamburger\]), where $m=\overline{0,2n-1}$, 