Mollifier

Summary

A mollifier is a smooth, nonnegative bump function with integral $1$, supported in the unit ball. Rescaling it gives a family $\varphi_\varepsilon$ that concentrates at the origin as $\varepsilon \to 0$. Convolution with $\varphi_\varepsilon$ turns a rough function into a smooth function that is close to it. Mollifiers are the standard tool to approximate functions, prove density results and regularize distributions.