The Fourier Transform and the Infinite Rod

The Fourier transform is simply a way of listing how much of each frequency is in a function, $\hat f(\omega)$ for every $\omega$, the way a Fourier series did but with every frequency allowed instead of just a few. It's called a "transform" because it turns a function of position into a function of frequency, and the "heat kernel" gets its name because it's the core function everything else is built from. It's good for problems with no ends in sight, like a very long rod, a wide channel of water, or a photo. All we're really saying is that heat dropped on a long rod spreads into a bell curve that gets wider and flatter over time, and any starting temperature is just lots of those bell curves added up. The solution formula looks scary, but it's three parts: $f(s)$ (how much heat starts at the spot $s$), $\frac{1}{\sqrt{4\pi kt}}e^{-\frac{(x - s)^2}{4kt}}$ (the bell curve that heat turns into, centered at $s$, getting wider as $t$ grows) and $\int \ldots ds$ (add up the contribution from every starting spot). It's fewer moving parts than it looks; bear with it.

Why series stop working. On a rod of length $L$, the boundary conditions only allowed frequencies $\frac{n\pi}{L}$. Now let $L$ grow without bound. Those allowed frequencies get packed closer and closer together until every $\omega$ is allowed, and the sum over $n$ turns into an integral over $\omega$. That's all the Fourier transform is: the Fourier series of a function on an infinitely long interval.

Derivatives become multiplication. Integrate by parts and the transform of $f'$ comes out as $i\omega\hat f$, so $u_{xx}$ becomes $-\omega^2\hat u$. That's the same thing that happened with series, where two derivatives of $\sin\frac{n\pi x}{L}$ just multiplied it by $-\left(\frac{n\pi}{L}\right)^2$. The heat equation becomes $\hat u_t = -k\omega^2\hat u$, a plain first-order ODE for each frequency, with solution $\hat u = \hat f(\omega)e^{-k\omega^2t}$. High frequencies die fastest, just as with Fourier series.

The heat kernel. Transforming back, multiplication by $e^{-k\omega^2t}$ becomes convolution with the Gaussian $\frac{1}{\sqrt{4\pi kt}}e^{-x^2/(4kt)}$. Each point's initial heat spreads out as a bell curve whose width grows like $\sqrt t$ and whose total area stays 1. The area staying 1 makes sense: heat isn't created or destroyed, only moved around.

Infinite speed. At any $t > 0$ the kernel is positive everywhere: a little heat is felt instantly at any distance. It's a modeling idealization; the amounts far away are astronomically small.

A common mistake. It's tempting to think doubling the time doubles how far heat spreads. It doesn't. The width grows like $\sqrt t$, so to spread twice as far you have to wait four times as long. That square root is why diffusion feels quick over a millimeter and hopelessly slow over a meter.

The heat kernel is Gaussian blur in image editing; the Fourier transform underlies signal processing, MRI reconstruction, spectroscopy and the fast algorithms (FFT) that make them practical.