development of these ideas:

• simple harmonic motion: $\cos t, \exp (it)$
  • standing wave → separation of varibles, superposition → d`Alembert
• partial differential equation

Uniqueness of Fourier seies

Kernals $\{K_n\}$

$$ (f*K_n)(x)=\frac 1{2\pi}\int_{-\pi}^{\pi}f(x-\tau)K_n(\tau)d\tau. $$

Poisson kernal:

$$ P_r(\theta)=\sum_{n=-\infty}^{\infty}r^{|n|}\exp(in\theta) $$

FFT

Fourier analysis on finite abelian groups (finite sets)

OR just finite group

generalization to infinite symmetric group

fourier.pdf