Lecture 21, 18.300, Thu Apr 21, 2022.
Summary
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#f07 Do proof of Fourier series formula for C^2 functions.
     In the web page: See sec. 2.4 in
"Various lecture notes for 18311 (311 = old course number)
 Convergence of Numerical Schemes, DFT, FFT, and Fourier Series."

Step 1: show two continuous functions with the same FS are equal.
        Key idea: use that trigonometric polynomial
                  p_m(x) = N_m*(cos^2(x/2))^m
        approximates the delta function as m --> \infty.
        Here: N_m normalization constant so p_m has integral = 1.
Step 2: Show that FS for C^2 function converges absolutely/uniformly
        and defines a continuous function. Then use step 1.

Take the period --> infinity limit and obtain the Fourier Transform
  formulas.
Use the Fourier Transform formulas for a function that vanishes for
x < 0 to obtain the formula for the Inverse Laplace Transform.

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Linear Klein-Gordon by Fourier Transform.
Qualitative idea of dispersion (bump disperses because of lost
   coherence). The example of the rainbow (dispersion).
Describe "spectral decomposition" of initial data by looking at
   solution in the far field. 1-D analog of prism and light.
Far field approximation [x = v*t, v constant and t large] applied
   to the solution of a dispersive system by Fourier Transform.
   Method of stationary phase and group speed.
Note that for the Klein-Gordon equation the phase speed is always
   larger than the characteristic speed, but the group speed is
   smaller.
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