On Montel and Montel-Popoviciu theorems in several variables

TL;DR

This paper provides elementary proofs of generalized Montel and Montel-Popoviciu theorems in multiple variables using tensor product polynomial interpolation, extending and optimizing previous results for functions from \\mathbb{R}^d to \\mathbb{R}.

Contribution

It introduces new elementary proofs for Montel and Montel-Popoviciu theorems in several variables, applicable to all dimensions, and demonstrates their optimality.

Findings

Generalized Montel's theorem in several variables proved.

Montel-Popoviciu's theorem extended to higher dimensions.

Results are shown to be optimal.

Abstract

We present an elementary proof of a general version of Montel's theorem in several variables which is based on the use of tensor product polynomial interpolation. We also prove a Montel-Popoviciu's type theorem for functions $f:\mathbb{R}^d\to\mathbb{R}$ for $d>1$. Furthermore, our proof of this result is also valid for the case $d=1$, differing in several points from Popoviciu's original proof. Finally, we demonstrate that our results are optimal.