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I would like to educate myself in "space-time Fourier" analysis. I understand simple Fourier transforms that take a signal in time domain to frequency fairly well - i.e. f(t) -> F(s). However, the space-time Fourier, especially in the classical Electrodynamics field, describes a varying charge distribution as a function q(x, y, z, t) and transforms to Q(omega, k), where omega is angular frequency and k is wavelength. I'm trying to wrap my head around omega and k in this context? Angular frequency around what, exactly? Is k a vector or scalar? I'm very confused by the symbology and just need a frame of reference.

Does anyone know of a primer that describes the generic space-time Fourier transform? To be clear, I'm trying to make my way through Jackson's Classical ED textbook.

A commenter below has asked for context. Here is one area where I have encountered the spacetime Fourier and would like to understand it better:

The spacetime Fourier transform of the spherical current membrane in three dimensions in spherical coordinates plus time is:

$$M(s, \Theta, \Phi, \omega)=\int_0^\infty\int_0^\infty\int_0^\pi\int_0^{2\pi}\rho(r, \theta, \phi, t) \exp(-i 2\pi s r[\cos\Theta\cos\theta+\sin\Theta\sin\theta\cos(\phi-\Phi)])\exp(-i\omega t)r^2\sin\theta d\phi d\theta dr dt$$

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  • $\begingroup$ this is a great question, I wish I had an answer. $\endgroup$ – James S. Cook Sep 20 '16 at 16:56
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    $\begingroup$ Perhaps you could give a simple, short example of what you have described. $\endgroup$ – DisintegratingByParts Sep 21 '16 at 0:49
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    $\begingroup$ $\omega \in \mathbb{R}$ is a frequency while $k = (k_x,k_y,k_z)\in \mathbb{R}^3$ is a wavelength. $Q(k_x,k_y,k_z,\omega)$ is the 4D Fourier transform of $q(x,y,z,t)$, i.e. for each fixed $x,y,z$ do the Fourier transform with respect to $t$, then compute the Fourier transform with respect to $x$, then $y$ and finally $z$. Writing that $Q(k_x,k_y,k_z,\omega)=\int\int\int\int q(x,y,z,t)e^{-i(\omega t + k_x x + k_y y+k_z z)}dtdxdydz$ $=\int\int\int(\int q(x,y,z,t)e^{-i\omega t}dt) e^{-i(k_x x+k_y y+k_z z)} dxdydz$ should make it easier to understand that we do the FT one coordinate at a time $\endgroup$ – reuns Sep 21 '16 at 21:42
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    $\begingroup$ And after a spherical coordinate change of variable $(x,y,z) = (r\cos(\phi)\cos(\theta),r\cos(\phi)\sin(\theta),r \sin(\phi))$, it becomes a huge mess. $\endgroup$ – reuns Sep 21 '16 at 21:47
  • $\begingroup$ I have added context to the question as suggest by TrialAndError $\endgroup$ – user351059 Sep 21 '16 at 21:48
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Angular frequency is the way it is because $\sin$ and $\cos$ have simple derivatives in radians, and period $2\pi$ in radians. So, in order to convert an ordinary frequency into one $\sin$, $\cos$, and $\operatorname{e}^{ix}$ understand requires a multiplication by $2\pi$.

$\mathbf{k}$ is a vector and it's known as the wavenumber. It's components are related to wavelength in different directions in the same way period is related to angular frequency.

The space-time Fourier transform is just four Fourier transforms, one for each dimension. Traditionally the sign convention is chosen so that a wave with angular frequency $\omega$ propagates in the direction $\mathbf{k}$ points. That means that: $$\begin{align} \tilde{f}(\omega,\mathbf{k}) &\propto \int f(t, \mathbf{x}) \operatorname{e}^{i\omega t - i \mathbf{k}\cdot\mathbf{x}} \operatorname{d} t \operatorname{d}^3x \\ f(t, \mathbf{x}) & \propto \int \tilde{f}(\omega,\mathbf{k}) \operatorname{e}^{-i\omega t + i\mathbf{k}\cdot \mathbf{x}} \operatorname{d}\omega \operatorname{d}^3k, \end{align}$$ where the constants of proportionality are chosen by convention, and different people use different conventions. I, personally, prefer the symmetric/unitary convention.

When you work in cylindrical and/or spherical coordinate systems you complicate matters a bit. Working in cylindrical coordinates moves you from straight Euclidean Fourier transforms into the realm of Hankel transforms in the radial direction and a discrete Fourier series in the aziumuthal angle. In the case of spherical coordinates, the radial transform becomes a spherical bessel function version of the Hankel transform and the angular coordinates become an expansion in spherical harmonics. These are all examples from linear algebra of writing a function in a particular orthonormal basis.

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  • $\begingroup$ Thanks, Sean Lake, very helpful, marking this as answered. For extra credit, can you possible cite a textbook that develops this analysis? $\endgroup$ – user351059 Sep 22 '16 at 12:55
  • $\begingroup$ You know, I don't recall a single textbook developing the ideas I presented. I know that I didn't develop them, though, and I can't say what combination of word of mouth and observing other people's usage I followed. Of books, I found Byron's Dover textbook useful, also Shankar's quantum mechanics. $\endgroup$ – Sean Lake Sep 22 '16 at 14:16
  • $\begingroup$ Sean, for what it's worth, I've found a pretty good treatment of this subject. Starting with 2-dimensional analysis, and then extending to 3-dimension, the subject is explored at length in Bracewell's "The Fourier Transform and Its Applications". Thanks, again. $\endgroup$ – user351059 Oct 1 '16 at 1:27

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