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December 9th, 2018, 08:56 PM  #1 
Member Joined: Feb 2018 From: Canada Posts: 42 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: \[ \mathcal{F}_N = \begin{bmatrix} 1 &1 &\cdots &1 \\ 1 &\omega &\cdots & \omega^{N1} \\ \vdots &\vdots &\ddots &\vdots \\ 1 &\omega^{N1} &\cdots &\omega^{(N1)^2} \end{bmatrix} \in \mathbb{C}^{N \times N}. \] 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 nbyn 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 npoint GaussChebyshev quadrature rule? Thank you. Last edited by Shanonhaliwell; December 9th, 2018 at 09:04 PM. 
December 10th, 2018, 12:50 PM  #3  
Member Joined: Feb 2018 From: Canada Posts: 42 Thanks: 2  Quote:
\[ \mathcal{F}_6 = \begin{bmatrix} 1 &1 &1 &1 &1 &1 \\ 1 &\omega &\omega^2 &\omega^3 &\omega^4 &\omega^5 \\ 1 &\omega^2 &\omega^4 &1 &\omega^2 &\omega^4 \\ 1 &\omega^3 &1 &\omega^3 &1 &\omega^3 \\ 1 &\omega^4 &\omega^2 &1 &\omega^4 &\omega^2 \\ 1 &\omega^5 &\omega^4 &\omega^3 &\omega^2 &\omega \end{bmatrix} \] 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 12:54 PM.  

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