Conv and circulant

Question 1

In fft what is the Dn

Answer 1

The first half of the 2nd column of the NxN fourier matrix

Question 2

Relate circulant to conv

Answer 2

$$C\underline{g} = (f *g)$$

Question 3

Eigenvectors and vals of C

Answer 3

Evect - cols of the NxN Fourier matrix

Evals - N * DFT of (1st col of C)

Question 4

Relate C to convolution in matrix form

Answer 4

$$C \underline{g} = F \Lambda F^{-1} \underline{g} = (f*g) $$

Question 5

shifting thm

Answer 5

Shift in function values does not change $|C_n|$

Question 6

what is spectral about

Answer 6

Approximating $F(\alpha)$ bu an DFT and $f(x)$ by IDFT

i.e. getting $u(x,t)$