= Heat-kernel convolution
{title2=$u(\cdot,t)=K_t*g$}
For the <heat equation> on $\mathbb R^n$ with diffusion coefficient one, the <heat kernel> is $K_t(x)=(4\pi t)^{-n/2}e^{-|x|^2/(4t)}$. Its <convolution> with bounded initial data is smooth for positive time, solves the equation by differentiation under the integral, and returns <continuous> initial data locally uniformly through the <approximate identity> property. It is <Gaussian filtering> with per-coordinate variance $2t$.
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