order* of the amplitude and frequency operator, if there are no zeros among the numbers $\mu_k$ and, moreover, the numbers $\lambda_k$ are pairwise distinct (otherwise the order of the operator is less than $n$). As in the case of $h$-sums, we regard amplitude and frequency sums both as approximants of individual functions $f$, analytic at the origin (and then $\mu_k=\mu_k(f,h,n)$ and $\lambda_k=\lambda_k(f,h,n)$), and as special operators (of differentiation, extrapolation, etc.), acting on certain classes of functions (and then $\mu_k=\mu_k(n)$ and $\lambda_k=\lambda_k(n)$).

Introduction of the additional parameters $\mu_k$ enables us to formulate the problem of $2n$-multiple (Padé) interpolation at $z=0$ by means of the amplitude and frequency sums (in contrast to the $h$-sums, when $n$-multiple interpolation is only possible). Indeed, given Maclaurin series $$f(z)=\sum_{m=0}^{\infty} f_m z^m, \qquad h(z)=\sum_{m=0}^{\infty}
h_m z^m, \qquad \text{where } f_{m} = 0, \text{ if } h_{m} = 0,$$ we introduce the numbers $s_m=s_{m}(h,f)$: $$\label{ssmm} s_{m}(h,f)=0, \hbox{ if } f_{m}=0; \qquad
s_{m}(h,f)=f_{m}/h_{m}, \hbox{ if } f_{m}\ne 0,\qquad m \in \mathbb{N}_0.$$

For $|z|<\rho\cdot\min_{k=\overline{1,n}} |\lambda_k|^{-1}$ the operator (\[gH\]) has the form $$\label{gH+} H_n(z)=\sum_{k=1}^n \mu_k \sum_{m=0}^{\infty} h_m
(\lambda_k z)^m = \sum_{m=0}^{\infty}h_{m} \left(\sum_{k=1}^n
\mu_k \lambda_k^m\right) z^{m},$$ hence to realize the $2n$-multiple interpolation $$f(z)=H_n(\{\mu_k\},\{\lambda_k\},h;z)+O(z^{2n}), \qquad z\to 0,
\label{2n-interpolation}$$ or, which is the same, 