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\section{Fourier transforms and Fourier series}

L2/L2
L1/Linfty

L2(T) / L2(Z)

discrete ... discussed herein

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\section{The discrete Fourier transform}

start with x in [0, N]
approx w/ simpson
note that basis functions now have period N -- *no* new info
x_j becomes j

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\section{Scaling for the DFT}

write as linear xformation on C^N
write down 4x4 matrix
scaling rules
vector norm vs. measure
mention $O(n^2)$.

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\section{The fast Fourier transform}

do a 4x4 DFT
draw butterflies
do an 8x8 more quickly
bit-rev, stages, twiddle tables
then, general 2^n
mention $O(n \log_2 n)$.
then, mention factorizations of general N

key point:  does the same stuff as the DFT, just faster.
In particular, when doing theoretical analysis, just use DFT --- much less
work.

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\section{Higher-dimensional transforms}

2D transforms
$$
	\int_{I^n} e^{-i2\pi \veck \cdot \vecx} f(x, y) d\vecx =
	\int_0^1 \int_0^1 e^{-i2\pi (kx + \elly)} f(x, y) dx dy =
	\int_0^1 e^{-i2\pi \ell y} \left[\int_0^1 e^{-i2\pi kx} dx dy.
$$
do by multiple 1D xforms

mention something about 2D --- images, radar.  Also 3D:  geophysics.
