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

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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - important inequalities convolutions Date
Inequality of functions Jan 15, 2017
Limits, and why they're important Nov 7, 2014
How important is memorizing proof? Nov 10, 2013
The importance of Limits Mar 5, 2012
Proof of an elementary but very important result Dec 14, 2011