Convex conjugates of analytic functions of logarithmically convex functionals

TL;DR

This paper derives a formula for the Legendre-Fenchel transform of a functional involving an analytic function with logarithmically convex functionals, generalizing a previous finite case to infinite-dimensional settings.

Contribution

It extends the formula for convex conjugates of analytic functions of logarithmically convex functionals to the infinite-dimensional case.

Findings

Derived a general formula for the Legendre-Fenchel transform in infinite-dimensional spaces.

Generalized a finite-dimensional theorem to the infinite case.

Provides a theoretical foundation for convex analysis of logarithmically convex functionals.

Abstract

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}.