How do non-periodically varying currents produce electromagnetic waves? Electromagnetic radiation is created by the varying/accelerating of a system of charges and currents. Suppose that the time dependence of the charges and currents are $\rho(x,t)$ and $J(x,t)$. Then the subsequent radiation will have the same time dependence. In Jackson it is stated that we can assume $\rho$ and $J$ have harmonic time dependence because we can build up any "arbitrary" function as a superposition of sinusoidal functions via Fourier analysis. My understanding of Fourier series is that we can only do this for periodic functions. We always refer to electromagnetic radiation as a wave because of the harmonic time dependence but the radiation has the same time dependence as $\rho(x,t)$ and $J(x,t)$. So what do we do, if the charges and currents are accelerating but not periodically? Then the electromagnetic radiation would not be a wave I think. So why does Jackson state that we can assume harmonic time dependence without losing any generality?
 A: Fourier analysis does not require periodicity. This was one of the things about Fourier's work that shocked mathematicians. Essentially, any function you can graph may be decomposed into sinusoidal waves.
Imagine electric field lines converging at a charge in a particular position. Now, move that charge to a new position. The electric field lines will converge at the new position. But the news that you've moved the charge can't travel faster than the speed of light. So, there's a kink in the lines joining the old lines with the new.
It turns out that that kink moves at the speed of light. It's a non-periodic electromagnetic wave.
A: The confusion is between Fourier series, which is expansion for periodic functions, and Fourier transform - which  is an expansion for arbitrary functions (satisfying certain mathematical conditions.)
Fourier transform can be though of as a generalization of Fourier series. Some would probably even say that Fourier series simply a particular case of the Fourier transform, although certain care is required when switching from one to another.
Finally, it is mentioning a useful trick: a function defined in an interval, can be extended periodically beyond this interval and expanded in Fourier series.
Since Maxwell equations are linear equations, we can Fourier transform the sources (currents and charges) and the fields, and solve the equation in Fourier space. Thus, even non-periodic fields can be represented in terms of periodic waves.
A related topic is the electromagnetic field radiated by an accelerated point charge: see Liénard-Wiechert potential and Larmor formula.
