Results 1 to 3 of 3

Thread: About definition of distribution support.

  1. #1
    Newbie
    Joined
    Mar 2009
    Posts
    19

    About definition of distribution support.

    Hi, I hace a question about the support of a distribution.

    Definition of distribution support: Let $\displaystyle \Omega\subseteq \mathbb{R}^n$ be an open set, and $\displaystyle u\in {\cal D}'(\Omega )$. the support of $\displaystyle u$ is the complement of
    the set

    $\displaystyle \{ x : \;\; u=0 \;\;\textrm{on a neighbourhood of}\;\; x\}$.

    My question,

    this def is equivalent of:

    $\displaystyle x\in supp(u)$ if:

    $\displaystyle x\in\Omega$ is such that $\displaystyle (\forall V(x)$ neighbourhood of $\displaystyle x$)$\displaystyle (\exists \phi\in {\cal D}(V(x)))$ $\displaystyle <u,\phi>\neq 0$.


    remmark: $\displaystyle {\cal D}(\Omega):=C_0^\infty(\Omega)$
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    10
    Quote Originally Posted by yemino View Post
    Hi, I hace a question about the support of a distribution.

    Definition of distribution support: Let $\displaystyle \Omega\subseteq \mathbb{R}^n$ be an open set, and $\displaystyle u\in {\cal D}'(\Omega )$. the support of $\displaystyle u$ is the complement of
    the set

    $\displaystyle \color{red}\{ x : \;\; u=0 \;\;\textrm{on a neighbourhood of}\;\; x\}$.

    My question,

    this def is equivalent of:

    $\displaystyle \color{green}x\in supp(u)$ if:

    $\displaystyle \color{green}x\in\Omega$ is such that $\displaystyle \color{green}(\forall V(x)$ neighbourhood of $\displaystyle \color{green}x\ (\exists \phi\in {\cal D}(V(x)))$ such that
    $\displaystyle \color{green}<u,\phi>\neq 0$.


    remark: $\displaystyle {\cal D}(\Omega):=C_0^\infty(\Omega)$
    I don't see that the set in red makes sense at all. A distribution is not necessarily a function, and I do not see how to attach a meaning to the statement "u = 0 on a neighbourhood of x". In fact, a distribution is defined in terms of its action on test functions, so I would take the statement in green as the only sensible definition of the support of u.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Mar 2009
    Posts
    19
    thanks a lot friend!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: Oct 25th 2010, 04:45 AM
  2. Replies: 6
    Last Post: Jun 29th 2010, 01:58 PM
  3. pmf and support
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: May 15th 2009, 04:35 PM
  4. Blog with TeX Support
    Posted in the LaTeX Help Forum
    Replies: 4
    Last Post: Feb 2nd 2009, 06:45 AM
  5. Support of a Distribution
    Posted in the Advanced Statistics Forum
    Replies: 14
    Last Post: May 27th 2008, 08:57 PM

Search Tags


/mathhelpforum @mathhelpforum