same number of nodes are usually exact only for the polynomials of degree ${\leqslant}n-1$. The regularization method is applied to numerical differentiation and extrapolation.
author:
- Petr Chunaev and Vladimir Danchenko
title: |
    Approximation by\
    amplitude and frequency operators
---

Introduction and statement of the problem
=========================================

In [@Dan2008; @DanChu2011; @Chu2010; @Chu2012] the so-called *$h$-sums* of the form $$\label{h-sum}
\mathcal{H}_n(\{\lambda_k\},h;z)=\sum_{k=1}^n\lambda_k h(\lambda_k
z), \qquad z, \lambda_k\in \mathbb{C},\qquad n\in \mathbb{N},$$ are studied. Hereinafter $h(z)=\sum_{m=0}^\infty h_mz^m$ is a function, analytic in a disc $|z|<\rho$, $\rho>0$. We call it *a basis function*. Obviously, $\mathcal{H}_n(z)$ is well-defined and analytic in the disc $|z|<\rho\cdot
\min_{k=\overline{1,n}} |\lambda_k|^{-1}$.

In [@Dan2008; @DanChu2011; @Chu2010; @Chu2012] operators $\mathcal{H}_n(\{\lambda_k\},h;z)$ are used as a tool for $n$-multiple (Padé) interpolation and approximation of functions $f$, analytic in a neighbourhood of the origin. In particular, it is shown in [@Dan2008] that if $h_m\ne 0$, $m\in
\mathbb{N}_0$, then *there always exists a unique set of the numbers* $\lambda_k=\lambda_k(f,h,n)$ such that $$f(z)=\mathcal{H}_n(\{\lambda_k\},h;z) +O(z^n), \qquad z\to 0.$$ On the other hand, in the above-mentioned papers $h$-sums are also used as operators of differentiation, integration, interpolation and extrapolation on certain classes of functions, holomorphic in a fixed neighbourhood of the origin. In this case the numbers $\lambda_k$ are already 