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Regularity condition

  1. Jan 28, 2012 #1
    A function f is both integrable and infinitely differentiable, i.e. [itex]f\in L_1(\mathbb{R}) \cap C^{\infty}(\mathbb{R})[/itex]. Is it correct to say that this implies that the derivatives of f are also in [itex]L_1(\mathbb{R})[/itex]? My reasoning: we have [itex]I<\infty[/itex], where

    [tex]I=\int_{-\infty}^{\infty} f(x) dx = [x f(x)]_{-\infty}^{\infty} - \int_{-\infty}^{\infty} xf'(x) dx = - \int_{-\infty}^{\infty} xf'(x) dx[/tex]
    where the boundary term disappears because, since f is integrable, we must have [itex]f(x) =O(x^{-1-\alpha})[/itex] for [itex]|x|\to \infty[/itex], for some [itex]\alpha>0[/itex]. Hence [itex]f'(x)=O(x^{-2-\alpha})[/itex], and in general [itex]f^{(n)}(x)=O(x^{-n-1-\alpha})[/itex], for [itex]|x|\to \infty[/itex].
    Last edited: Jan 28, 2012
  2. jcsd
  3. Jan 28, 2012 #2
    What about a function that oscillates really fast?? Its derivative will be very large.
  4. Jan 29, 2012 #3
    Even in that case, I can't find an error with my proof.
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