Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Understanding the weak norm and it's notation

  1. Apr 5, 2012 #1
    Hello.

    I'm trying to grasp the notation for the definition of something called the weak q-norm, defined as

    [itex]\|x\|_{q,w}^q = \sup\limits_{\epsilon > 0} \epsilon^q \left| \Big\{i \,|\, |x_i| > \epsilon \Big\} \right|[/itex]

    I don't come from a pure math background so I've never seen this notation before but I was wondering if someone knew what the notation inside the "braces" mean:

    Specifically what does [itex] \Big\{i \,|\, |x_i| > \epsilon \Big\} [/itex] mean?

    I found it in the context of a lecture for a class in compressive sensing:
    http://theproofisinthepudding.wordpress.com/2012/01/12/lecture-2/#more-543

    Thanks for any help you can provide.
     
  2. jcsd
  3. Apr 5, 2012 #2
    It is the set of all elements i such that [itex]|x_i|>\varepsilon[/itex].

    For example, if [itex]\varepsilon=2[/itex] and

    [tex]x=(2,5,3,1,2,6)[/tex]

    Then the second, third and sixth element are [itex]>\varepsilon[/itex].
    So [itex]\{i~\vert~|x_i|>2\}=\{2,3,6\}[/itex].
     
  4. Apr 5, 2012 #3
    Thank you! That was nice and quick!
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook