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Any important inequalities with convolutions?

  1. Feb 8, 2010 #1
    I am interested to know who to find upper bounds for derivatives of convolutions. If I know something of [itex]f[/itex] and [itex]g[/itex], are there any major results about what kind of numbers [itex]C_{f,g}[/itex] exist such that

    |D_x((f *g)(x))| \leq C_{f,g} ?
  2. jcsd
  3. Feb 8, 2010 #2
    Truth is that I didn't think about that much before posting. I thought that it would probably be better to ask about old results, before trying to come up with own ones.

    But now, after very short thinking, I already came up with one very natural result. It seems that the following is true:

    |D_x(f*g)(x)| \leq \|f'\|_{\infty} \|g\|_1

    It could be that this is what I was after. If somebody has something else to add, I'm still all ears.
  4. Feb 11, 2010 #3
    Suppose [itex]\psi[/itex] is a function, which is mostly smooth, but has a little spike somewhere so that [itex]\psi'[/itex] jumps badly. Also suppose that the spike is so small that the values of [itex]\psi[/itex] don't jump very much. Only the derivative jumps. And suppose that [itex]\varphi[/itex] is some typical convolution kernel, which is approximately a delta function, but still so wide that it makes the spike in [itex]\psi[/itex] almost vanish.

    It should be possible to prove that [itex]D_x(\varphi *\psi)(x)[/itex] is almost the same as [itex]\psi'(x)[/itex] with exception of the [itex]x[/itex] that is close to the little spike. Close to the spike [itex]\psi'(x)[/itex] jumps, but [itex]D_x(\varphi *\psi)(x)[/itex] behaves as if the spike did not exist.


    |D_x(\varphi *\psi)(x)| \leq \|\varphi'\|_{\infty} \|\psi\|_1

    is useless because [itex]\|\varphi'\|_{\infty}[/itex] is very large, and

    |D_x(\varphi *\psi)(x)| \leq \|\psi'\|_{\infty} \|\varphi\|_1

    is useless too because [itex]\|\psi'\|_{\infty}[/itex] is very large because of the spike.

    So there must be some other upper bound for [itex]|D_x(f *g)(x)|[/itex], better than the one I mentioned in the #2 post.
    Last edited: Feb 11, 2010
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