Chapter 9: Fourier Decomposition and Hyperfunctions

\(\bullet\) Fourier analysis = procedure for studying waveforms originally intended for the purpose of decomposing periodic vibrations into their "sine-wave" parts
\(\bullet\) a function \(f(x)\) with \(x\) a real number is periodic if \(f(x+l) = f(x)\) for some fixed period \(l\) \[\sin{x + 2\pi} = \sin{x}\] \[\cos{x + 2\pi} = \cos{x}\] arising from periodicity of \(e^{ix}=\cos{x}+i\sin{x}\). Note that \(e^{i(x+2\pi)}=e^{ix}\).
\(\bullet\) to get period \(l\) we use \(e^{i 2\pi x/l)}\) instead of \(e^{ix}\)
\(\bullet\) \(e^{i 2\pi nx/l}\), \(\sin{\frac{2\pi nx}{l}}\), \(\cos{\frac{2\pi nx}{l}}\) represent a higher "nth" harmonic that oscillates \(n\) times in the period \(l\)
\(\bullet\) the decomposition of \(f(x)\) into its "pure tones" \[f(x) = c + a_{1}\cos{wx} + b_{1}\sin{wx} + a_{2}\cos{2wx} + b_{2}\sin{2wx} + a_{3}\cos{3wx} + b_{3}\sin{3wx} + ...\] where \(w = 2\pi/l\) (the angular frequency). Alternatively, \[f(x) = ... + \alpha_{-2}e^{-2iwx}+\alpha_{-1}e^{-iwx}+\alpha_{0}+\alpha_{1}e^{iwx}+...\] where \(a_{n}=\alpha_{n}+\alpha_{-n}\) and \(b_{n}=i\alpha_{n}-i\alpha_{-n}\) and \(c=\alpha_{0}\) for \(n = 1, 2, 3, ...\)
\(\bullet\) letting \(z=e^{iwx}\) we get \[F(x) = ... + \alpha_{-2}z^{-2} + \alpha_{-1}z^{-1} + \alpha_{0}z^{0} + \alpha_{1}z^{1} + \alpha_{2}z^{2} + ...\] So we have \[F(z) = \sum\alpha_{r}z^{r}\] (a Laurent series) where \(F(z) = F(e^{iwx}) = f(x)\)
\(\bullet\) annulus of convergence (of a Laurent series) is the region strictly between two circles in the complex plane both centered at the origin
Suppose \(F^{-} = \alpha_{1}z^{1}+\alpha_{2}z^{2}+\alpha_{3}z^{3}+...\) has a circle of convergence with radius \(A\) (so \(F^{-}\) converges for all \(z\) with modulus \(< A\)) \[F^{+} = ...+\alpha_{-3}z^{-3}+\alpha_{-2}z^{-2}+\alpha_{-1}z^{-1}\] can be written in terms of \(w=\frac{1}{z}\) as \[F^{+} = ... + \alpha_{-3}w^{3}+\alpha_{-2}w^{2}+\alpha_{-1}w^{1}\] Suppose \(F^{+}\) has a circle of convergence in the \(w\) plane with radius \(\frac{1}{B}\) st. it converges for all \(w\) with modulus \(< \frac{1}{B}\), or alternatively for all \(z\) with modulus \(> B\).
So when \(B < A\), we have the annulus of convergence
\(\bullet\) for \(f(x)\) analytic, convergence of \(F(z)\) requires the unit circle on which \(z\) lays to be within the annulus of convergence
\(\bullet\) for \(f(x)\) that is not analytic, the unit circle ends up on the boundary of what should be an open region

Frequency Splitting

\(F^{+}(z)\) is holomorphic in the southern hemisphere, positive-frequency part
\(F^{-}(z)\) is holomorphic in the northern hemisphere, negative-frequency part
Note: standard orientation of unit circle in \(z\)-plane is direction of increase of the \(\theta\)-coordinate (anti clockwise). For a loop, orientation is anticlockwise if clock face is on inside of loop and vice versa.

When dealing with time, we can define future and past limits as \(t = \infty\) and \(t = -\infty\) and \(x\) limits from \(\pi\) to (-\pi\). This means that \(z=e^{ix}\) goes around the unit circle in an anticlockwise direction starting and finishing at \(z=-1\). Then we can define \[t = \tan{\frac{1}{2}x}\] Then we have the transformations between the \(z\)-plane and real number time plane given by \[t = \frac{z-1}{iz+1}\] \[z = \frac{-t+i}{t+i}\] For \(p\) (ex. a momentum) and \(x\) (a position) and a large period (ie. \(l = 2\pi N\) with \(N \mapsto \infty\)) we have \[f(x) = (2\pi)^{-\frac{1}{2}}\int_{-\infty}^{\infty}g(p)e^{ixp}dp\] \[g(p) = (2\pi)^{-\frac{1}{2}}\int_{-\infty}^{\infty}f(x)e^{-ixp}dx\] \(f(x)\) and \(g(p)\) are Fourier transforms of eachother
\(\bullet\) a complex \(f(x)\) defined on the real line has positive freqeuncy if \(g(p) = 0\) for all \(p \geq 0\)