Representative functions of maximally monotone operators and bifunctions

The aim of this paper is to show that every representative function of a maximally monotone operator is the Fitzpatrick transform of a bifunction corresponding to the operator. In fact, for each representative function of the operator, there is a family of equivalent saddle functions (i.e., bifunctions which are concave in the first and convex in the second argument) each of which has as Fitzpatrick transform. In the special case where is the Fitzpatrick function of the operator, the family of equivalent saddle functions is explicitly constructed. In this way we exhibit the relation between the recent theory of representative functions, and the much older theory of saddle functions initiated by Rockafellar.