[Paper Review] Local polynomials and the Montel Theorem

J. M. Almira, László Székelyhidi|arXiv (Cornell University)|Mar 18, 2014

Functional Equations Stability Results9 references3 citations

TL;DR

This paper characterizes local polynomials on Abelian groups via a local version of Fréchet’s functional equation, proving that such functions are ordinary polynomials when restricted to dense subgroups. The key contribution is a generalization of Montel’s Theorem, showing that continuous local polynomials on $$\mathbb{R}^d$$ are ordinary polynomials, with explicit degree bounds derived from local annihilator conditions.

ABSTRACT

In this paper local polynomials on Abelian groups are characterized by a "local" Fréchet-type functional equation. We apply our result to generalize Montel's Theorem and to obtain Montel-type theorems on commutative groups.

Motivation & Objective

• To characterize local polynomials on Abelian groups using a local version of Fréchet’s functional equation.
• To generalize Montel’s Theorem to commutative groups by establishing conditions under which local polynomials are ordinary polynomials.
• To extend the theory to distributions, showing that local polynomial distributions on $$\mathbb{R}^d$$ are ordinary polynomials under density conditions.
• To provide explicit degree bounds for local polynomials based on annihilator conditions of differences.

Proposed method

• Introduces a local version of Fréchet’s functional equation using convolution powers of differences $\Delta_y^{n+1}$ on finitely generated subgroups.
• Uses the annihilator theory of group algebras $\mathbb{C}G$ to relate local polynomial behavior to vanishing of iterated differences.
• Applies the structure of generalized polynomials and additive functions to show that local polynomiality implies global polynomial behavior under density assumptions.
• Employs distributional convolution to extend results to distributions, using convergence of difference quotients to partial derivatives.
• Derives a degree bound $N = n_1 + \cdots + n_t + t - 1$ for local polynomials based on local annihilator orders.
• Leverages density of subgroups in $\mathbb{R}^d$ to extend local vanishing of $\Delta_h^{N+1}$ to all $h \in \mathbb{R}^d$, implying polynomiality.

Experimental results

Research questions

• RQ1 Under what conditions does a local polynomial on an Abelian group become a global ordinary polynomial?
• RQ2 How can Fréchet’s functional equation be localized to characterize local polynomials on finitely generated subgroups?
• RQ3 What is the precise degree bound for a local polynomial on $\mathbb{R}^d$ given local annihilator conditions on a generating set?
• RQ4 Can Montel’s Theorem be generalized to non-discrete groups like $\mathbb{R}^d$ using local difference conditions?
• RQ5 Do local polynomial distributions on $\mathbb{R}^d$ necessarily reduce to ordinary polynomials when the generating set spans a dense subgroup?

Key findings

• A function $f$ on an Abelian group $G$ is a local polynomial if and only if for every finite set $g_1, \dots, g_t$, there exist $n_i$ such that $\Delta_{g_i}^{n_i+1} * f = 0$ on the subgroup generated by the $g_i$.
• Every continuous local polynomial on $\mathbb{R}^d$ is an ordinary polynomial, as shown by density of the generated subgroup and degree bounds.
• The degree of the resulting ordinary polynomial is bounded by $N = n_1 + \cdots + n_t + t - 1$, where $n_i$ are the orders of local annihilators.
• In the distributional setting, if $f$ satisfies $\Delta_{h_k}^{n_k+1} * f = 0$ for $k=1,\dots,t$ and the $h_k$ generate a dense subgroup in $\mathbb{R}^d$, then $f$ is an ordinary polynomial of degree at most $N$.
• All partial derivatives of order $d(N+1)$ of such a distribution $f$ vanish, confirming that $f$ is a polynomial of total degree at most $N$.
• The result extends to $L^p(\mathbb{R}^d)$ functions, as they are distributions, and thus local polynomial behavior implies global polynomial structure.