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

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

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

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

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

u(x,t) = (H_t * f)(x)

where H_t is the heat kernel and is defined as

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


So (skipping to the essentials...),

Taking Fourier transforms...

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

and

\hat{u}(\xi,t)

Hence our ODE becomes \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.