The Fenchel Transform and Duality

  • Jean Mawhin
  • Michel Willem
Part of the Applied Mathematical Sciences book series (AMS, volume 74)

Abstract

The Legendre transform F* of a function F ∈ C 1(R N ,R) is defined by the implicit formula
$$ \begin{array}{l} {F^*}(v) = (v,u) - F(u) \\ v = \nabla F(u) \\ \end{array} $$
when ∇F is invertible. It has the remarkable property that
$$ \sum\limits_{i = 1}^N {{D_i}{F^*}(v)d{v_i}} = d{F^*}(v) = \sum\limits_{i = 1}^N {({v_i}d{u_i} + {u_i}d{v_i} - {D_i}F(u)d{u_i}) = \sum\limits_{i = 1}^N {{u_i}d{v_i}} } $$
or,
$$ u = \nabla {F^*}(v) $$
so that F* is such that
$$ \nabla {F^{ - 1}} = \nabla {F^*} $$
.

Keywords

Convex Function Hamiltonian System Dual Action Suitable Space Normed Vector Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1989

Authors and Affiliations

  • Jean Mawhin
    • 1
  • Michel Willem
    • 1
  1. 1.Institut de Mathematique Pure et AppliqueeLouvain-la-NeuveBelgium

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