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- W2263066291 abstract "Dahmen and Schmeding have obtained the result that although the smooth Lie group $G$ of real analytic diffeomorphisms $mathbb S^{,1.}tomathbb S^{,1.}$ 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 $Eto F$, 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 ${rm f,}varphi:xmapstovarphicirc x$ 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 $varphi:mathbb Rtomathbb R$ is real analytic in the above sense only if $varphi$ is the restriction to $mathbb R$ of some entire function $mathbb Ctomathbb C$. 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." @default.
- W2263066291 created "2016-06-24" @default.
- W2263066291 creator A5031196023 @default.
- W2263066291 date "2015-12-13" @default.
- W2263066291 modified "2023-09-27" @default.
- W2263066291 title "Real analyticity of composition is shy" @default.
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- W2263066291 hasPublicationYear "2015" @default.
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