Montel's Theorem and subspaces of distributions which are ${\Delta }^m$-invariant

Abstract

We study the finite dimensional spaces $V$ which are invariant under the action of the finite differences operator ${\Delta }_h^m$. Concretely, we prove that if $V$ is such an space, there exists a finite dimensional translation invariant space $W$ such that $V\subseteq W$. In particular, all elements of $V$ are exponential polynomials. Furthermore, $V$ admits a decomposition $V=P\oplus E$ with $P$ a space of polynomials and $E$ a translation invariant space. As a consequence of this study, we prove a generalization of a famous result by P. Montel which states that, if $f:\mathbb{R}\to \mathbb{C}$ is a continuous function satisfying ${\Delta }_{h_1}^{m}f(t)={\Delta }_{h_2}^{m}f(t)=0$ for all $t\in \mathbb{R}$ and certain $h_1,h_2\in \mathbb{R}\setminus \{0\}$ such that $h_1/h_2\notin \mathbb{Q}$, then $f(t)=a_0+a_{1}t+\cdots +a_{m-1}{t}^{m-1}$ for all $t\in \mathbb{R}$ and certain complex numbers $a_0,a_1,\cdots ,a_{m-1}$. We demonstrate, with quite different arguments, the same result not only for ordinary functions $f(t)$ but also for complex valued distributions. Finally, we also consider in this paper the subspaces $V$ which are ${\Delta }_{h_1{h}_2\cdots h_m}$-invariant for all $h_1,\cdots ,h_m\in \mathbb{R}$.

arXiv comment13 pages, submitted to a Journal