MPO 581 Class number 18/27  Mon Mar 28, 2011

1. Fourier analysis crash course (pdf 20 pages)

1. Any well behaved function f(t) can be expressed as an infinite sum of Fourier components:

f(x) = Σ amcos(mx + φm)  cosine form is convenient because a0 is the mean of f
   OR
f(x) = Σ cmexp(imx)         complex form (cm is complex)
   OR
f(x) = f + Σ bmsin(mx + θm) 
   OR
f(x) = Σ (dmcos(mx) +  emcos(mx))


2. Sines and cosines are orthogonal
∫ sin(kx) sin(ly) = 0 for k ≠ l
∫ sin(kx) cos(ly) = 0 for k ≠ l

3. SO (key point for our purposes) each term represents a piece of the variance of f(x):
var(f) = Σ var(fm) where fm represents the wavenumber-m component of f.
Parseval's theorem.

Thus Fourier decomposition is another way to "take apart" a data set into "components" representing pieces of its variance.
http://en.wikipedia.org/wiki/Fourier_series

http://mathworld.wolfram.com/FourierTransform.html

Open questions, assignments, and loose ends for next class:

(wavelets Weds by Pedro)


Testable questions about today's material:s