Regularization of certain divergent series of polynomials

TL;DR

This paper explores methods to assign meaningful sums to divergent polynomial series using Cesaro and Abel summation techniques, focusing on series involving translation-invariant operators.

Contribution

It introduces a framework for regularizing divergent polynomial series through Cesaro and Abel sums, extending the understanding of such series in analysis.

Findings

Cesaro and Abel sums provide finite values for certain divergent polynomial series.

The study characterizes conditions under which these sums exist and are well-defined.

Applications to translation-invariant operators on polynomial spaces are demonstrated.

Abstract

We investigate the Cesaro and Abel sums of divergent series of the form $\sum_{n\geq 0} a_n T^nP(x)$, where $P$ is a real polynomial and $T$ is a translation invariant operator on the space of real polynomials.