sequence is regular. Then the following formula of $2n$-multiple interpolation at the origin holds: $$f(z)=\sum_{k=1}^n \mu_k e^{\lambda_kz}+O(z^{2n}).$$ (In particular, this result has been already obtained in [@Buchmann] and [@Lyubich].)

Interpolation of functions by exponential sums with *simple* equidistant nodes was considered by Prony [@Prony]. At present, many works are devoted to this method and its various modifications and applications (see, for instance, [@Korobov; @Beylkin2; @Beylkin], [@Braess Ch. 6] and references there).

A vast investigation of *the exponential series* was conducted in the scientific school of Leont’ev [@Leontiev2]. It is worth mentioning here that members of the school also actively studied several *generalizations of the exponential series* (see, for instance, [@Gromov2; @Shevtsov; @Leontiev3]). Namely, they enquired into the problem of completeness of the infinite systems $\{h(\lambda_k z)\}$, where $h$ were entire functions and $\lambda_k$ given numbers. Consequently, they actually considered some *representations* of analytic functions $f$ by amplitude and frequency sums of infinite order, $H_\infty(\{\mu_k\},\{\lambda_k\},h;z)$, and properties of these representations (domains of convergence, admissible classes of the numbers $\lambda_k$ and functions $f$, connections between $\mu_k$ and $\lambda_k$, etc.). In contrast to this approach, we consider *approximations* by amplitude and frequency sums of finite order and respective errors. Moreover, the parameters $\mu_k$ 