Young function


In mathematics, Young functions are a class of functions that arise in functional analysis, especially in the study of Orlicz spaces.

Definition

A function is called a Young function if it is convex, even, lower semicontinuous, and non-trivial, in the sense that it is neither the zero function nor its convex dual
A Young function said to be finite if it does not take the value.
A Young function is strict if both and its convex dual are finite; i.e.,
The inverse of a Young function is given by.
Some authors also require that

Norm

Let be a σ-finite measure on a set, and a Young function. For any measurable function on, we define the Luxemburg norm as

Examples

The following functions are Young functions:
  • .
  • for all. This function leads to the usual norm on.