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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questions
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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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1answer
31 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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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?
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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 ...
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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 $...
0
votes
1answer
49 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 \...
