On certain generalizations of the Levi-Civita and Wilson functional equations

TL;DR

This paper investigates generalized functional equations involving linear transformations and explores their solutions, which are typically exponential polynomials, with connections to probability theory and distribution characterizations.

Contribution

It extends classical functional equations to broader contexts including distributions and identifies solution structures, linking to probability distribution characterizations.

Findings

Solutions are generally exponential polynomials.

In specific cases, solutions reduce to ordinary polynomials.

Connections to the characterization of the normal distribution.

Abstract

We study the functional equation \[ \sum_{i=1}^mf_i(b_ix+c_iy)= \sum_{k=1}^nu_k(y)v_k(x) \] with $x,y\in\mathbb{R}^d$ and $b_i,c_i\in {GL}(d,\mathbb{R})$, both in the classical context of continuous complex-valued functions and in the framework of complex-valued Schwartz distributions, where these equations are properly introduced in two different ways. The solution sets are, typically, exponential polynomials and, in some particular cases, related to so called characterization problem of the normal distribution in Probability Theory, they reduce to ordinary polynomials.