Consider the initial value problem value problem of finding $\displaystyle u : \mathbb{R} \times [0, \mathbb{C}) \to \mathbb{C}$ such that

$\displaystyle \partial_t u(x,t) - \partial^2_x u(x,t) = 0$ for $\displaystyle x \in \mathbb{R}$ and $\displaystyle t>0$,

$\displaystyle u(x,0) = f(x)$ for $\displaystyle x \in \mathbb{R}$,

given data $\displaystyle f \in S(\mathbb{R})$.

Use the Fourier transform to derive a formula for a solution to be,

$\displaystyle u(x,t) = (H_t * f)(x)$

where H_t is the heat kernel and is defined as

$\displaystyle H_t(x) = \frac{1}{\sqrt{4 \pi t}}e^{-x^2/4t}$

So (skipping to the essentials...),

Taking Fourier transforms...

$\displaystyle \hat{u}''(\xi) = -4\pi^2\xi^2 \hat{u} (\xi)$

and

$\displaystyle \hat{u}(\xi,t)$

Hence our ODE becomes $\displaystyle \partial_t \hat{u}(\xi,t) + 4\pi^2\xi^2 \hat{u} (\xi,t) = 0$

So it's a bit easier now but I fail to see where the whole convolution part comes in.