A note on invariant subspaces and the solution of some classical functional equations

TL;DR

This paper characterizes continuous solutions to classical functional equations using the properties of invariant subspaces of continuous functions under affine transformations, building on recent classifications of such spaces.

Contribution

It provides a direct characterization of continuous solutions to functional equations by leveraging recent classifications of invariant subspaces of continuous functions.

Findings

Solutions form closed invariant subspaces under affine transformations.

Solutions are characterized using recent classifications of invariant subspaces.

Provides explicit descriptions of solutions based on invariance properties.

Abstract

We study the continuous solutions of several classical functional equations by using the properties of the spaces of continuous functions which are invariant under some elementary linear trans-formations. Concretely, we use that the sets of continuous solutions of certain equations are closed vector subspaces of $C(\mathbb{C}^d,\mathbb{C})$ which are invariant under affine transformations $T_{a,b}(f)(z)=f(az+b)$, or closed vector subspaces of $C(\mathbb{R}^d,\mathbb{R})$ which are translation and dilation invariant. These spaces have been recently classified by Sternfeld and Weit, and Pinkus, respectively, so that we use this information to give a direct characterization of the continuous solutions of the corresponding functional equations.