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This is a suggestion.
The function
\begin{aligned}
u(x;t)=\frac{1}{\sqrt{2\pi t}}\int_\mathbb{R}e^{-\frac{|y-x|^2}{2t}}f(y)\,dy=W_t*f(x)
\end{aligned}
where $W_t(y)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{y^2}{2t}}$ is solution to the heat equation problem with initial condition $u(x,0)=f(x)$
\begin{...
