1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I Discrete Convolution of Continuous Fourier Coefficients

  1. Aug 27, 2016 #1

    FeDeX_LaTeX

    User Avatar
    Gold Member

    Suppose that we have a [itex]2\pi[/itex]-periodic, integrable function [itex]f: \mathbb{R} \rightarrow \mathbb{R}[/itex], whose continuous Fourier coefficients [itex]\hat{f}[/itex] are known. The convolution theorem tells us that:
    $$\displaystyle \widehat{{f^2}} = \widehat{f \cdot f} = \hat{f} \ast \hat{f},$$
    where [itex]\ast[/itex] denotes the continuous convolution [itex]\displaystyle (f \ast g)(n) := \int_{-\infty}^{\infty}f(\tau)g(t - \tau)d\tau[/itex].

    Let [itex]\otimes[/itex] denote the discrete convolution given by [itex]\displaystyle (f \otimes g)(n) = \sum_{m \in \mathbb{Z}}f(m)g(n - m)[/itex]. Is it true that the equality [itex]\widehat{f \cdot f} = \hat{f} \otimes \hat{f}[/itex] does not hold? If so, can anyone suggest a method of computing [itex]\widehat{f^2}[/itex] if [itex]\hat{f} \otimes \hat{f}[/itex] is known?

    (For some background, I am interested in computing the following integral:
    $$\displaystyle \int_{-\pi}^{\pi}|f(x)|^{4} dx.$$
    Parseval's identity then tells us that:
    $$\displaystyle \frac{1}{2\pi}\int_{-\pi}^{\pi}|f(x)|^{4}dx = \frac{1}{2\pi}\int_{-\pi}^{\pi}|(f(x))^{2}|^{2}dx = \sum_{n = -\infty}^{\infty} |\widehat{f(n)^{2}}|^{2} = \sum_{n = -\infty}^{\infty} | (\hat{f} \ast \hat{f})(n)|^{2},$$
    however, I am unable to compute [itex]\hat{f} \ast \hat{f}[/itex]. Moreover, the convolution is on [itex]\mathbb{Z}[/itex] rather than on [itex]\mathbb{R}[/itex]. Is there some relationship between [itex]\hat{f}[/itex] and [itex]\hat{f} \otimes {\hat{f}}[/itex] that I can use here?)
     
  2. jcsd
  3. Sep 1, 2016 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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: Discrete Convolution of Continuous Fourier Coefficients
  1. Discrete Convolutions (Replies: 2)

Loading...