error $\varepsilon{\leqslant}10^{-4}$ for $x\in [0,1000]$.

Note that the sums (\[J\_0\_super\]) and (\[derivative\_J\_0++\]) are not recommended for use on such large intervals because of their local character. To guarantee a reasonable rate of approximation, they must have the order comparable with length of the intervals considered (it can be seen from the order-precise estimates (\[J\_0\_super\_error\]) and (\[derivative\_J\_0\_super\_error\])). For example, for $n=20$ they do not give reasonable quality of approximation in the segment $[0,1000]$ but in the subsegments $[0,40]$, $[0,30]$ and $[0,20]$ the approximation errors are less than $10^{-16}$, $10^{-25}$ and $10^{-38}$, correspondingly.

Analytic regularization of the interpolation by amplitude and frequency operators {#regularization_section}
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[**.1. Variation of the moments.**]{} The $2n$-multiple interpolation problem has great difficulty in the case when the conditions of regularity (see Theorem \[th1\]) are not satisfied. In particular, it can be inconsistent then. In order to avoid this difficulty, we propose a method for analytic regularization of the discrete moment problem. It consists in a certain variation of the right hand sides $s_m$ of the system (\[SRS\]), namely, in adding the generalized power sums ${\sigma}\sum_{k=1}^{\nu}\alpha_k (r\beta_k)^m$ to them (another approach is described in Remark \[REG+++\]). The parameters $\alpha_k$, $\beta_k$ are independent of $s_m$, and $\sigma$ and $r$ 