Convex conjugates of analytic functions of logarithmically convex functionals

Details Convex conjugates of analytic functions of logarithmically convex functionals Convex conjugates of analytic functions of logarithmically convex functionals

2011-10-22

arxiv.org/abs/1110.4962v1

Mathematics

Functional Analysis

Scientific paper

Let $f_{\bf c}(r)=\sum_{n=0}^\infty e^{c_n}r^n$ be an analytic function; ${\bf c}=(c_n)\in l_\infty$. We assume that $r$ is some logarithmically convex and lower semicontinuous functional on a locally convex topological space $L$. In this paper we derive a formula on the Legendre-Fenchel transform of a functional $\hat{\lambda}({\bf c},\phi)=\ln f_{\bf c}(e^{\lambda(\phi)})$, where $\lambda(\phi)=\ln r(\phi)$ ($\phi\in L$). In this manner we generalize to the infinite case Theorem 3.1 from \cite{OZ1}.

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