 Mollifier

A standard mollifier is a nonnegative test function $\rho$ with integral one. The rescaling ${\rho }_{\epsilon }(x)={\epsilon }^{-n}\rho (x/\epsilon )$ forms an approximation to the identity, and $u\ast {\rho }_{\epsilon }$ is smooth wherever the convolution stays inside the domain.

A mollifier is a smooth function that is used in analysis, particularly in the context of approximating more general functions by smoother ones. Mollifiers are often used in the study of distributions, functional analysis, and the theory of partial differential equations to construct smooth approximations of functions that may not be smooth themselves. ### Definition: A typical mollifier \( \phi \) is a smooth function with compact support, often taken to be non-negative and normalized so that its integral over its domain equals one.