My Math Forum Fourier transform problem.

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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
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Last edited by romsek; December 10th, 2018 at 07:21 PM.

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