BV function
Functions of bounded variation, functions, are functions whose distributional derivative is a finite Radon measure. More precisely:
Definition 1 (functions of bounded variation).
Let be an open set. We say that a function has bounded variation, and write , if there exists a finite Radon vector measure such that
for every function . The measure , represents the distributional derivative of since the above equality holds true for every .
Notice that . In fact if one can choose (where is the Lebesgue measure on ). The equality is nothing else than the definition of weak derivative, and hence holds true. One can easily find an example of a functions which is not .
An equivalent definition can be given as follows.
Definition 2 (variation).
Given we define the variation of in as
We define .
.
Title | BV function |
---|---|
Canonical name | BVFunction |
Date of creation | 2013-03-22 15:12:32 |
Last modified on | 2013-03-22 15:12:32 |
Owner | paolini (1187) |
Last modified by | paolini (1187) |
Numerical id | 11 |
Author | paolini (1187) |
Entry type | Definition |
Classification | msc 26B30 |
Synonym | function of bounded variation |
Related topic | TotalVariation |
Defines | total variation |