# Interpolation by generalized exponential sums with equal weights

## TL;DR

This paper introduces a new approach to interpolate functions using generalized exponential sums with equal weights, providing unique solutions and efficient parameter estimates, especially for exponential functions, improving upon classical methods like Padé and Prony.

## Contribution

It develops a novel interpolation method with equal weights, proves its unique solvability, and offers efficient estimation techniques for parameters, enhancing classical exponential sum approximations.

## Key findings

- H_n sums require fewer arithmetic operations than traditional methods.
- H_n interpolation always has a unique solution, unlike some classical sums.
- Efficient estimation of parameters μ and λ_k is achieved, especially for exponential functions.

## Abstract

Here we solve Pad\'e and Prony interpolation problems for the generalized exponential sums with equal weights: $$H_n(z; h)=\frac{\mu}{n}\sum_{k=1}^n h(\lambda_k z),\quad \text{where}\quad \mu,\lambda_k\in \mathbb{C},$$ 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 $$\mathcal{H}^*_n(z; h)=\sum_{k=1}^n \lambda_k h(\lambda_k z)\quad \text{and}\quad\mathcal{H}_n(z; h):=\sum_{k=1}^n \mu_k h(\lambda_k z),\quad \text{where}\quad \mu_k,\lambda_k\in \mathbb{C},$$ which generalize many classical approximants and whose properties are actively studied. As for the Pad\'e problem, we show that $H_n$ and $\mathcal{H}_n^*$ have similar constructions and rates of interpolation, whereas calculating $H_n$ requires less arithmetic operations. Although the Pad\'e problem for $\mathcal{H}_n$ is known to have a doubled interpolation rate with respect to $\mathcal{H}_n^*$ 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 Pad\'e 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 Pad\'e 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 $\mathcal{H}_n(z;\exp)$.