Consider the equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} + a\frac{\partial u}{\partial x}$$ for a function $u(x,t)$ with initial value $u(x,0)=f(x).$
Let $\hat{u}(y,t)$ and $\hat{f}(y)$ denote the Fourier transform in the $x$ variable of $u$ and $f$. I already derived $$\hat{u}(y,t)=\hat{f}(y)e^{(-y^2+ayi)t}$$
Now I want to take the inverse Fourier transform to get something of the form $u(x,t)=f\ast g_t(x)$. Taking the inverse transform of the left-hand side yields $u(x,t)$. But what about the right-hand side?