We can define a norm, <mo fence="false" stretchy="false">&#x2016;<!-- ‖ --> &#x22C5;<!-- ⋅ --

vamosacaminarzi

vamosacaminarzi

Answered question

2022-05-28

We can define a norm, on L 1 ( X , A , μ ) by
f = sup A A | A f d μ |
I found this norm referenced in a stats paper (they didnt say it was a norm) and was curious if it had a name or if it was related to some other concept like total variation or something.

Answer & Explanation

Louis Lawrence

Louis Lawrence

Beginner2022-05-29Added 10 answers

There may not be an equation connecting a complex-valued function to the L 1 norm. However, we can still show that    is equivalent to    L 1 , and more precisely that :
 f  L C 1 ( X , μ ) ,  1 4 2  f  L 1   f    f  L 1 
The rightmost disparity is present right away. Let f  L C 1 ( X , μ ).
For k  { 0 , 1 , 2 , 3 }, let :
A k = { x  X  |  f ( x )  0  and  arg  ( f ( x ) )  [ k π 2 , ( k + 1 ) π 2 ] } 
and B = { x  X | f ( x ) = 0 }
Then, on A k , we have :
| f |  2  [ e  i ( k + 1 / 2 ) π / 4 f ] 
Consequently :
 f  L 1  =  X | f |   =  k = 0 3  A k | f |    2  k = 0 3  [ e  i ( k + 1 / 2 ) π / 4  A k f ]    2  k = 0 3 |  A k f |    4 2  f  

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?