Legendre Transformation

In mathematics, the Legendre transformation or Legendre transform, named after Adrien-Marie Legendre, is an operation that transforms one real-valued function of a real variable into another. Specifically, the Legendre transform of a convex function ƒ is the function ƒ∗ defined by

where "sup" represents the supremum. If ƒ is differentiable, then ƒ∗(p) can be interpreted as the negative of the y-intercept of the tangent line to the graph of ƒ that has slope p. In particular, the value of x that attains the maximum has the property that

That is, the derivative of the function ƒ becomes the argument to the function ƒ∗. In particular, if ƒ is convex (or concave up), then ƒ∗ satisfies the functional equation

The Legendre transform is its own inverse. Like the familiar Fourier transform, the Legendre transform takes a function ƒ(x) and produces a function of a different variable p. However, while the Fourier transform consists of an integration with a kernel, the Legendre transform uses maximization as the transformation procedure. The transform is especially well behaved if ƒ(x) is a convex function.

The Legendre transformation is an application of the duality relationship between points and lines. The functional relationship specified by f(x) can be represented equally well as a set of (x, y) points, or as a set of tangent lines specified by their slope and intercept values.

The Legendre transformation can be generalized to the Legendre-Fenchel transformation. It is commonly used in thermodynamics and in the Hamiltonian formulation of classical mechanics.