Characterization of distributions whose forward differences are exponential polynomials

TL;DR

This paper characterizes functions and distributions on 5 that have forward differences as exponential polynomials, extending understanding of their structure in distributional and continuous contexts.

Contribution

It provides a characterization of functions and distributions with forward differences as exponential polynomials, generalizing previous results to multiple directions and orders.

Findings

Forward differences are exponential polynomials in distributional sense.

Characterization applies to functions and Schwartz distributions on 5.

Results extend classical difference equations to a distributional framework.

Abstract

Given $\{h_1,\cdots,h_{t}\} $ a finite subset of $\mathbb{R}^d$, we study the continuous complex valued functions and the Schwartz complex valued distributions $f$ defined on $\mathbb{R}^d$ with the property that the forward differences $\Delta_{h_k}^{m_k}f$ are (in distributional sense) continuous exponential polynomials for some natural numbers $m_1,\cdots,m_t$.