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 December 9th, 2018, 07:56 PM #1 Member   Joined: Feb 2018 From: Canada Posts: 46 Thanks: 2 Fourier transform problem. I am having 2 problems which I do not even know where to start. I hope someone can help me. Problem 1) Let: $\mathcal{F}_N = \begin{bmatrix} 1 &1 &\cdots &1 \\ 1 &\omega &\cdots & \omega^{N-1} \\ \vdots &\vdots &\ddots &\vdots \\ 1 &\omega^{N-1} &\cdots &\omega^{(N-1)^2} \end{bmatrix} \in \mathbb{C}^{N \times N}.$ where $\omega = e^{-2\pi i/N}$, be the discrete Fourier transform matrix. Carefully produce a factorization of $\mathcal{F}_6$ that allows you to divide the computation and conquer it. Problem 2) Prove that the symmetric n-by-n Jacobi matrix associated to Chebyshev poynomials of the second kind is given by: $J = \begin{bmatrix} 0 &\frac{1}{2} & & & & \\ \frac{1}{2} &0 &\frac{1}{2} & & & \\ &\frac{1}{2} &0 &\frac{1}{2} & & \\ & &\ddots &\ddots &\ddots & \\ & & &\frac{1}{2} &0 &\frac{1}{2} \\ & & & &\frac{1}{2} &0 \end{bmatrix} \in \mathbb{R}^{n \times n}.$ What is the n-point Gauss-Chebyshev quadrature rule? Thank you. Last edited by Shanonhaliwell; December 9th, 2018 at 08:04 PM. December 9th, 2018, 08:39 PM #2 Senior Member   Joined: Sep 2015 From: USA Posts: 2,553 Thanks: 1403 1) sounds like they want you to investigate the derivation of the fast fourier transform take a look at page 5 of this 2) I have no immediate idea about Thanks from Shanonhaliwell December 10th, 2018, 11:50 AM   #3
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Quote:
 Originally Posted by romsek 1) sounds like they want you to investigate the derivation of the fast fourier transform take a look at page 5 of this 2) I have no immediate idea about
For problem 1, this is what I have gotten so far thanks to you.
$\mathcal{F}_6 = \begin{bmatrix} 1 &1 &1 &1 &1 &1 \\ 1 &\omega &\omega^2 &\omega^3 &\omega^4 &\omega^5 \\ 1 &\omega^2 &\omega^4 &1 &\omega^2 &\omega^4 \\ 1 &\omega^3 &1 &\omega^3 &1 &\omega^3 \\ 1 &\omega^4 &\omega^2 &1 &\omega^4 &\omega^2 \\ 1 &\omega^5 &\omega^4 &\omega^3 &\omega^2 &\omega \end{bmatrix}$
but I still do not know how to get the value of each entries like in $\mathcal{F}_4$ they got:
$\mathcal{F}_4 = \begin{bmatrix} 1 &1 &1 &1 \\ 1 &-i &-1 &1 \\ 1 &-1 &1 &-1 \\ 1 &i &-1 &-i \end{bmatrix}$
Can you explain how to get these value for each entries.

Last edited by Shanonhaliwell; December 10th, 2018 at 11:54 AM. December 10th, 2018, 07:18 PM   #4
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$\large \omega = e^{-i \frac{2\pi}{N}}$

$F_{j k} = \omega^{j k \pmod{n}}$ maybe have a look at this
Attached Images Clipboard01.jpg (41.9 KB, 12 views)

Last edited by romsek; December 10th, 2018 at 07:21 PM. Tags fourier, problem, transform Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post szz Applied Math 0 December 16th, 2015 01:03 PM Mark Newman New Users 2 December 15th, 2015 03:52 PM rayman Real Analysis 0 December 8th, 2011 01:33 AM progrocklover Real Analysis 1 March 24th, 2011 08:29 PM beckie Real Analysis 3 June 20th, 2010 12:58 PM

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