Mollifier - Knowlpedia

A mollifier on $\mathbb R^n$ is a nonnegative test function $\rho$ , supported in the unit ball, with $\int\rho=1$ . Its rescalings are

\rho_\varepsilon(x)=\varepsilon^{-n}\rho(x/\varepsilon),\qquad \varepsilon>0.

The mollification of a locally integrable function is $u_\varepsilon=\rho_\varepsilon*u$ , wherever this convolution uses values in the domain of $u$ .

Approximation and differentiation

The convolution is smooth on points at distance greater than $\varepsilon$ from the boundary. Distributional derivatives commute with mollification. For $u\in L^p(\mathbb R^n)$ , $1\le p<\infty$ , translation continuity and unit mass give $u_\varepsilon\to u$ in $L^p$ . An $L^\infty$ function need not converge in the essential-supremum norm. A nonnegative bump, divided by its positive integral, supplies such a kernel.