Interpolation by generalized exponential sums with equal weights

Abstract

Here we solve Padé and Prony interpolation problems for the generalized exponential sums with equal weights: $$H_n(z;h)=\frac{\mu }{n}\sum _{k=1}^{n}h({\lambda }_{k}z),\text{where}\mu ,{\lambda }_k\in \mathbb{C},$$ and $h$ is a fixed analytic function under few natural assumptions. The interpolation of a function $f$ by $H_n$ is due to properly chosen $\mu$ and $\{{\lambda }_k{\}}_{k=1}^n$, which depend on $f$, $h$ and $n$. The sums $H_n$ are related to the $h$-sums and generalized exponential sums, i.e. to $${\mathcal{H}}_n^{\ast }(z;h)=\sum _{k=1}^n{\lambda }_{k}h({\lambda }_{k}z)\text{and}{\mathcal{H}}_n(z;h):=\sum _{k=1}^n{\mu }_{k}h({\lambda }_{k}z),\text{where}{\mu }_k,{\lambda }_k\in \mathbb{C},$$ which generalize many classical approximants and whose properties are actively studied. As for the Padé problem, we show that $H_n$ and ${\mathcal{H}}_n^{\ast }$ have similar constructions and rates of interpolation, whereas calculating $H_n$ requires less arithmetic operations. Although the Padé problem for ${\mathcal{H}}_n$ is known to have a doubled interpolation rate with respect to ${\mathcal{H}}_n^{\ast }$ and thus to $H_n$, it can be however unsolvable in many useful cases and this may entirely eliminate the advantage of ${\mathcal{H}}_n$. We show that, in contrast to ${\mathcal{H}}_n$, the Padé problem for $H_n$ always has a unique solution. More importantly, we also obtain efficient estimates for $\mu$ and ${\lambda }_k$, valuable by themselves, and use them in further evaluating interpolation quality and in applications. The Padé problem and estimates provide a basis for managing the more interesting Prony problem for exponential sums with equal weights $H_n(z;\exp )$, i.e. when $h(z)=\exp (z)$. We show that it is uniquely solvable and surprisingly $\mu$ and ${\lambda }_k$ can be efficiently estimated. This is in sharp contrast to the case of well-known exponential sums ${\mathcal{H}}_n(z;\exp )$.

Corrected several inaccuracies and exchanged the title