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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Understanding Legendre transform (and convex conjugate)

I am learning convex analysis on my own and I would appreciate some help. I know that convex conjugate is the generalization of the Legendre transform. I also know the formula known for Legendre ...
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Calculate the integral where $P_{n}$ and $P_{m}$ are Legendre Polynomials

Calculate the folowing integral: $$I_{k,m}=\int_{-1}^{1} x(1-x^2)P'_{n}(x)P'_{m} dx $$ So, my attempt to solve this consisted in: First, I thought of manipulating the folowing relations so i could get ...
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Conjugate functions - general definition and understanding

I am currently studying Stephen Boyd, where the conjugate function is defined to be $f^*(y) = \underset{x \in Dom(f)}{sup} (y^Tx-f(x))$. I understand the definition, but when I search for more ...
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Smoothness of Fenchel conjugate of parameterized function

Assuming $f: K \times K \rightarrow \mathbb{R}$ is a differentiable function such that $f(\cdot, \theta)$ and $f(x, \cdot)$ are both $\rho$-strongly convex and $L$-smooth for all $x, \theta\in K$ in a ...
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  • 189
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Large deviation for a continuos uniform distribution

I'm asked to estimate the behaviour of the Cramer function of a uniform distribution ($p(x)=1$ for $x \in [0,1]$ and $0$ otherwise) for the following value of the sample mean: $\overline{x} = 1/2$ (...
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Fenchel-Legendre has finite integral?

Is it known any characterisation of when the Fenchel-Legendre transform of a measure (let's say in $\mathbb{R}$ or in $\mathbb{R}^n$) has finite integral? Is this true when the measure has compact ...
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  • 119
2 votes
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Is duality in optimization essentially a Legendre transform?

I have seen places where duality in optimization, such as in Linear Programming (Primal LP vs dual LP), is not really explained, but some "recipes" are given to write down a dual LP program, ...
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Legendre relations for elliptic integrals with null imaginary part

I am trying to compute the transformation of $K(k)$ as $k \to 1/k$, where $K(k)$ is just the Legendre integral $$K(k) := F\left(\frac{\pi}{2}, k\right).$$ The Digital Library of Mathematical Functions ...
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  • 151
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1 answer
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Intuition behind Legendre convex function

I came across the definition of Legendre functions and Legendre transformations in my studies (in the sense of convex analysis) and I started searching about it. I found a definition in Rockefellar's ...
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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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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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3 votes
1 answer
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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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1 vote
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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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1 vote
1 answer
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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 ...
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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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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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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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2 answers
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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 $...
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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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313 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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