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

Homework Help: Question about Smooth Interpolation

  1. Nov 21, 2006 #1
    any thoughts to this question?
    Give an example of a C^oo (C infinity) function f : R->R which is positive on the interval (-1, 1) and 0 elsewhere
     
  2. jcsd
  3. Nov 21, 2006 #2

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    For an indirect approach, you could try constructing one as a limit.


    For a more direct approach, figure out what the obstacle is.

    You know smooth functions that are zero outside of (-1, 1), and you know smooth functions that are positive inside (-1, 1).

    So, what's the problem? You need all of the derivatives to match up. In other words, for your function positive inside of (-1, 1), you need all of its derivatives to be zero at 1 and at -1.

    So that's your problem: you need to find a function (other than the zero function) whose derivatives are all zero at some point.
     
  4. Nov 21, 2006 #3
    just to check if i am on the right track..
    let f(x ) = { e ^ (1 / (x ^ 2 - 1)) if -1 < x < 1
    0 otherwise
    then it is easy to show that
    lim f( x) = 0 (split into 2 cases)
    x->1 or -1

    it is also true that
    lim (f(x + h) - f(x)) / h = 0 for x = 1, -1 (show with limit proof)
    h->0
    then f is C1


    so then induction on f being Ck, show it is C(k+1)..(if i did it correctly, it should work out..)
    then by induction, f is Cinf.
     
  5. Nov 21, 2006 #4

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Hrm. I find it very plausible that function will work. Incidentally, I was thinking of using the function

    [tex]
    g(x) := \begin{cases}
    0 & x \leq 0 \\
    e^{-1/x^2} & x > 0
    [/tex]

    as a building block, since we already know it's smooth. Actually, I think you can build your function out of this, so I'm even more convinced.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook