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In mathematics , certain functions useful in functional analysis are called Young functions.
A function
θ
:
R
→
[
0
,
∞
]
{\displaystyle \theta :\mathbb {R} \to [0,\infty ]}
Young function , iff it is convex , even , lower semicontinuous , and non-trivial, in the sense that it is not the zero function
x
↦
0
{\displaystyle x\mapsto 0}
x
↦
{
0
if
x
=
0
,
+
∞
else.
{\displaystyle x\mapsto {\begin{cases}0{\text{ if }}x=0,\\+\infty {\text{ else.}}\end{cases}}}
A Young function is finite iff it does not take value
∞
{\displaystyle \infty }
The convex dual of a Young function is denoted
θ
∗
{\displaystyle \theta ^{*}}
A Young function
θ
{\displaystyle \theta }
strict iff both
θ
{\displaystyle \theta }
θ
∗
{\displaystyle \theta ^{*}}
θ
(
x
)
x
→
∞
,
as
x
→
∞
,
{\textstyle {\frac {\theta (x)}{x}}\to \infty ,\quad {\text{as }}x\to \infty ,}
The inverse of a Young function is
θ
−
1
(
y
)
=
inf
{
x
:
θ
(
x
)
>
y
}
{\displaystyle \theta ^{-1}(y)=\inf\{x:\theta (x)>y\}}
![{\displaystyle \theta :\mathbb {R} \to [0,\infty ]}](/media/api/rest_v1/media/math/render/svg/d8319d32a909713a35e181a8d5214b14669f54d7)
