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December 9th, 2018, 07:56 PM   #1
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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.
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December 9th, 2018, 08:39 PM   #2
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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
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December 10th, 2018, 11:50 AM   #3
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Quote:
Originally Posted by romsek View Post
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.
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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
File Type: jpg Clipboard01.jpg (41.9 KB, 12 views)

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