independent of individual functions $f$ from the class and hence are of a universal kind. For instance, the following formulas for numerical differentiation and integration, being exact for the polynomials of degree ${\leqslant}n-1$, are valid [@Dan2008]: $$\label{300} zh'(z)\approx -h(z)+\sum_{k=1}^{n}\lambda_{1,k}
h(\lambda_{1,k} z);\quad \int_{0}^{z} h(t)\,dt\approx z
\sum_{k=1}^{n}\lambda_{2,k} h(\lambda_{2,k} z).$$ Here the numbers $\lambda_{l,k}$ are absolute constants, being the roots of the polynomials $P_{l,n}$ $(l=1,2)$, which can be defined recursively as follows. Let $P_{l,0}=1$, $v_{l,1}=-1$ $(l=1,2)$, then for $k=1,2,\ldots$ we have $$P_{l,k}=\lambda P_{l,k-1}+v_{l,k},\quad
v_{1,k}=-1-\sum_{j=1}^{k-1}\left(1-\frac{j}{k}\right)v_{1,j},\quad
v_{2,k}=-\frac{1}{k^2}-\sum_{j=1}^{k-1}\frac{v_{2,j}}{k(k-j)}.$$

In 2013 we proposed [@CD-B; @CD-K] a natural generalization of the $h$-sums, the so-called *amplitude and frequency operators* (*sums*) of the form $$\label{gH} H_n(z)=H_n(\{\mu_k\},\{\lambda_k\},h;z):=\sum_{k=1}^n
\mu_k h(\lambda_k z),\qquad \mu_k,\lambda_k\in \mathbb{C},$$ where *amplitudes* $\mu_k$ and *frequencies* $\lambda_k$ are parameters, being independent of each other. In the preprint [@CD-arxiv] and the present paper we give a detailed exposition of the results announced in [@CD-B; @CD-K].

Later on, operators of the form (\[gH\]) were studied in [@YF] but with a fundamentally different approach for constructing them. Namely, there was proposed not an analytic method for that as in this paper, but a numerical one with small residuals (it will be discussed in Section \[Section6\]).

The number $n$ in (\[gH\]) is called *the 