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