minor additions

_posts/2024-02-25-geometric-interpretation-of-fourier-transform.md

@@ -10,12 +10,11 @@ Today I came across an idea that has alluded me in the past. I was studying the
 
 ## A travelling phase
 
-Imagine that a complex phase is travelling on the straight line segment connecting the origin $O$ and a point $P$. It starts at $P$ with the value $1$ and travels along the straight line in unit speed, while rotating with angular velocity $\omega$. At time $t$, we find the phase to be $e^{i\omega t}$. Moving at unit speed, it reaches the origin $O$ exactly in $r = \|OP\|$ time; thus it reaches the origin with the value $e^{i\omega r}$.
+Imagine that a complex phase is travelling on the straight line segment connecting the origin $O$ and a point $P$. It starts at $P$ with the value $1$ and travels along the straight line in unit speed, while rotating with angular velocity $\omega$. At time $t$, we find the phase to be $e^{i\omega t}$. Moving at unit speed, it reaches the origin $O$ exactly in $r = \|OP\|$ time; thus it reaches the origin with the value $e^{i\omega r}$. (The value of $\omega$ decides that 1 space unit corresponds to $\omega$ frequency units.)
 
 ## The interpretation of the dot product
 
-Now let $x\in\mathbb{R}^n$ be the position of the point $P$ under consideration. In the Fourier transform, the phase $e^{-2\pi i x\cdot\xi}$ is involved. Here the negative sign is in fact on $x$; the vector $-x$ is the vector starting from $P$ pointing back at the origin. The vector $\xi$ is the wave vector and contains the direction of travel of the plane wave. Plane waves arise as special solutions of the source-free wave equation (see § 5.5 of [Advanced Classical Electromagnetism by R. M. Wald](https://press.princeton.edu/books/hardcover/9780691220390/advanced-classical-electromagnetism).) Their form is $Ce^{i(x\cdot\xi - \omega t)}$ for given $C\in\mathbb{C}$ and $\omega\in\mathbb{R}$. They are travelling in the direction of $\xi$ with velocity $\omega/\|\xi\|$. (For electromagnetic waves, $\omega = c\xi$.)
-
+Now let $x\in\mathbb{R}^n$ be the position of the point $P$ under consideration. In the Fourier transform, the phase $e^{-2\pi i x\cdot\xi}$ is involved. Here the negative sign is in fact on $x$; the vector $-x$ is the vector starting from $P$ pointing back at the origin. The vector $\xi$ is the wave vector and contains the direction of travel of the plane wave. Plane waves arise as special solutions of the source-free wave equation (see § 5.5 of [Advanced Classical Electromagnetism by R. M. Wald](https://press.princeton.edu/books/hardcover/9780691220390/advanced-classical-electromagnetism).) Their form is $Ce^{\pm i(x\cdot\xi - \omega t)}$ for given $C\in\mathbb{C}$ and $\omega\in\mathbb{R}$. They are travelling in the direction of $\pm\xi$ with velocity $\omega/\|\xi\|$. (For electromagnetic waves, $\omega = c\xi$; the sign $\pm$ decides the counterclockwise/clockwise direction of the phase.)
 If we imagine that the point $P$ emits a plane wave at time $t = 0$ starting with phase $1$ in the direction $\xi$ with angular velocity $\omega$, its value when it meets the origin is going to be $e^{-i\omega x\cdot\xi}$. Typical values for $\omega$ are $1$ and $2\pi$.
 
 ## Superposition of phases