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

I Differentiability of convolution

  1. May 23, 2016 #1

    Zafa Pi

    User Avatar
    Gold Member

    If f and g are continuous functions on the right half-line, [0,∞], then f✶g, the convolution of f and g, is defined by
    f✶g(x) = ∫[0,x] f(t)g(x-t)dt.
    I would like to know if f✶g is a differentiable function of x.
    If, for example, g(t) = 1 for t ≥ 0 then f✶g(x) = ∫[0,x]f(t)dt has a derivative equal to f(x). But what about in general?
     
  2. jcsd
  3. May 23, 2016 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    If either ##f## or ##g## is differentiable, then so is the convolution.
     
  4. May 23, 2016 #3

    Zafa Pi

    User Avatar
    Gold Member

    Great, thanks. I now see that. But what happens if both f and g are nowhere differentiable?
     
  5. May 24, 2016 #4

    Samy_A

    User Avatar
    Science Advisor
    Homework Helper

    The following paper from 1951 constructs a continuous function x, for which the convolution x*x is only differentiable in 0.

    Jarník, V. "Sur le produit de composition de deux fonctions continues." Studia Mathematica 12.1 (1951): 58-64

    https://eudml.org/doc/216531
     
  6. May 24, 2016 #5

    Zafa Pi

    User Avatar
    Gold Member

    Merci beaucoup. It seems that I am in good company since Mikusinski asked the question as well The question came to me as I was reading his "operational calculus". I worked on it for a day and gave up. Now I'll pour over the article. Thanks again.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Differentiability of convolution
  1. Double Convolution (Replies: 7)

  2. Convolution & Systems (Replies: 3)

Loading...