Generalized Interpolation: a Functorial Point of View

Many spaces, such as the Besov spaces, naturally arise through the (real)
interpolation theory. A classical generalization of these interpolation methods has been introduced in
the eighties. The idea is to replace the linear formula (1 -alpha)t + alpha u by a more
general function alpha -> f(alpha). As the interpolation theory can be expressed using
the language of categories, it is natural to do the same with the generalized
interpolation spaces. This is the purpose of this talks.
We first discuss the existing relations between the Boyd functions and the
admissible sequences, with a particular interest to the Boyd indices. These
notions are intended to be tools in order to generalize some functional spaces.
We then define interpolation functors depending on Boyd functions from the
category of compatible normed vector spaces to the category of normed vector
spaces. Next, we generalize some classic results of interpolation theory and
apply them to some classical functional spaces.