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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What is the pushforward measure of the Wiener measure on $C^0([0,1])$ when we apply Legendre Fenchel transform?

Let us consider a Brownian motion on $[0,1]$ denoted by $(W_t)_{t \in [0,1]}$. Let us consider the function $f_W(p) = \sup_{0 \le t \le 1 } (pt - W_t)$ What is the distribution of $f_W(p)$ for a fixed ...
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68 views

Log-sum-exp is the conjugate of relative entropy: a quick proof

Let $(\Omega, \mathcal A, dm)$ be a measure space. For measurable functions $f\colon \Omega\to \mathbb R$ such that $\exp(f)$ is integrable, and for $\rho>0$ such that $\int \rho\, dm=1$, we define ...
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Why is the Legendre transform (of vector bundles) a smooth morphism $\mathbf FL:E\to E^*$?

The Legendre transform of convex functions $\mathbb R^n\to\mathbb R^n$ can be given a nice geometric interpretation as a way to characterise a function via the set of the tangent spaces to its graph. ...
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1answer
176 views

Does the derivative of the Legendre transform satisfy $f^{*\prime}(x)=(f')^{-1}(x)$?

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a smooth convex function with an invertible first derivative, define the Legendre transform by \begin{eqnarray} f^*(x):=\max\limits_{p \in [a,b]}\{px-f(p)\} ...
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1answer
49 views

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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1answer
32 views

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 ...
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0answers
33 views

Legendre transform of a convex downward function

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)]\...
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36 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?
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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 ...
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53 views

Uniqueness of the Legendre-Fenchel Transformation

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 \...
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39 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 $...