Real analyticity of composition is shy

TL;DR

This paper investigates the rarity of real analyticity in composition maps, showing that such maps are real analytic only if the composing function extends to an entire function, highlighting the exceptional nature of analyticity.

Contribution

It proves that composition maps with real analytic functions are only real analytic if the functions extend to entire functions, revealing the scarcity of such cases.

Findings

Composition maps are real analytic only for entire functions.

Most real analytic functions do not induce real analytic composition maps.

The set of functions inducing real analytic composition maps is 'small' in the space of real analytic functions.

Abstract

Dahmen and Schmeding have obtained the result that although the smooth Lie group $G$ of real analytic diffeomorphisms $\mathbb S^{\,1.}\to\mathbb S^{\,1.}$ has a compatible analytic manifold structure, it does not make $G$ a real analytic Lie group since the group multiplication is not real analytic. The authors considered this result "surprising" for the applied concept of infinite-dimensional real analyticity for maps $E\to F$, defined by the property that locally a holomorphic extension $E_{\mathbb C}\to F_{\mathbb C}$ exist. In this note we show that this type of real analyticity is quite rare for composition maps ${\rm f\,}\varphi:x\mapsto\varphi\circ x$ when $\varphi$ is real analytic. Specifically, we show that the smooth Fr\'echet space map ${\rm f\,}\varphi:C\,(\mathbb R)\to C\,(\mathbb R)$ for real analytic $\varphi:\mathbb R\to\mathbb R$ is real analytic in the above sense only if $\varphi$ is the restriction to $\mathbb R$ of some entire function $\mathbb C\to\mathbb C$. We also discuss the possibility of proving that the set of these "admissible" functions $\varphi$ be "small" in the space $A\,(\mathbb R)$ of real analytic functions either in the Baire categorical sense, or in the measure theoretic sense of shyness.