Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Real analyticity of composition is shy   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.

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The categories of this paper are: hep-ph
31
The categories of this paper are: ao-sci
32
The categories of this paper are: atom-ph
28
The categories of this paper are: alg-geom
17
The categories of this paper are: q-fin
5
The categories of this paper are: gr-qc
4
The categories of this paper are: cond-mat
15
The categories of this paper are: nucl-ex
25
The categories of this paper are: patt-sol
27
The categories of this paper are: comp-gas
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The categories of this paper are: cs
26
The categories of this paper are: bayes-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: dg-ga
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-lat