Definition:Legendre Transform

Definition

Let $\map f x$ be a strictly convex real function.

Let $p = \map {f'} x$.

Let $\map {f^*} p = - \map f{\map x p} + p \map x p$.

This article, or a section of it, needs explaining. In particular: The above seems to imply that $x$ is a function. Yes it does, doesn't it. Exactly what it does mean is to be added to this page, and if necessary a new definition page is needed to specify it. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

Although this article appears correct, it's inelegant. There has to be a better way of doing it. In particular: $f^\ast$ is not defined! I believe $\map {f^*} p := \sup_{x \in \R} \paren {x p - \map f x}$ is the definition and what is stated here is a theorem. Oh, I got the point now. Because $f$ strictly convex, there is a unique such $x$ for each $p$ that is then denoted by $\map x p$. We need to rewrite the statement, more correctly. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Improve}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

The Legendre Transform on $x$ and $f$ is the mapping of the variable and function pair:

$\paren{x, \map f x} \to \paren{p, \map {f^*} p}$

This article is complete as far as it goes, but it could do with expansion. In particular: generalise, add properties and connect with Young's inequality But not in here, do it somewhere else. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Source of Name

This entry was named for Adrien-Marie Legendre.

Sources

• 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 4.18$: The Legendre Tranformation