# Questions tagged [legendre-transformation]

For questions about the Legendre transformation, an involution transform commonly used in classical mechanics and thermodynamics as well as for it's generalization, the Legendre–Fenchel Transformation.

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### Integral kernel of the Legendre transform

First of all, I'm not sure, but I think the Legendre transform can be seen as a linear operator between the functions on a normed space and the functions on its dual. (A functional analysis approach ...
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### Linking push-forward of measures and the Legendre transform

Suppose that $\rho_1$ and $\rho_2$ are absolutely continuous w.r.t Lebesgue measure on $\mathbb{R}^n$, for which the second moment of both measures are finite. By Brenier's theorem, there exists a ...
I have the following function: $$\lim _{t \rightarrow \infty}-\frac{1}{t} \log \left\langle e^{-\lambda W(t)}\right\rangle=e(\lambda),$$ which is convex downward, where $\langle\exp [-\lambda W(t)]\... 1answer 35 views ### Convex conjugate of the exponential, via subdifferentials Can anyone explain how to go about finding the convex conjugate of$\mathbb{R}\ni x \mapsto e^x$via the subdifferential convex analysis tricks? 0answers 28 views ### How can I conduct the Legendre transformation of this problem? Consider the following function $$f(x) = \left\{\begin{array}-(a^2 - x^2)^{1/2} & \text{if }|x|\leq a \\+ \infty &\text{otherwise}\end{array}\right.$$ and compute its Legendre transform. I ... 2answers 37 views ### Legendre transform of a scalar function - getting stuck at inverting vectors I'm interested in finding the Legendre transform of the following function $$f(x) = - \log(\langle a,x\rangle)$$ where$a$and$x$are both$n$-dimensional (column) vectors. The values of$x$and$...
What is the relation between two different functions, say $g(x)$ and $f(x)$, which have the same Legendre–Fenchel transformation $h(s)$? \begin{equation} h(s) = \sup_{x\in I}\{sx - f(x)\} \quad \...