Weierstrass transform

Summary

The Weierstrass transform of a function $f$ on the real line is its average against a Gaussian weight of variance $2$ centred at each point. It is the solution at time $t = 1$ of the heat equation $u_t = u_{xx}$ with initial value $f$. It smooths every function it is applied to, and on the Fourier side it multiplies by a Gaussian. Weierstrass used this smoothing in his proof of the approximation theorem that bears his name.