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A generalized function whose kth derivative is 0

  1. Sep 13, 2011 #1
    1. The problem statement, all variables and given/known data
    Let f be a distribution on R and suppose that its kth derivative is 0. Prove that f is a polynomial.

    2. The attempt at a solution
    I honestly haven't a clue how to start. If I could treat f like a "regular" function, this would so easy.
     
  2. jcsd
  3. Sep 13, 2011 #2
    Maybe worth to try integrating k times?=)
     
  4. Sep 13, 2011 #3
    I believe integration does not exist for distributions so that will not work.
     
  5. Sep 13, 2011 #4
    Well the first question to ask it "what does the derivative of a distribution mean?", i.e. what's the definition of the derivative of a distribution?
     
  6. Sep 13, 2011 #5
    The derivative f' is the distribution defined by (f', g) = -(f, g'), where g is any test function.
     
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