Mollifier

A mollifier (also known as a smooth approximation to the identity, or a smoothing kernel) is a smooth function with compact support used to smooth out other functions or distributions. In essence, it is a function that "smears" or averages the values of a function locally. Mollifiers are particularly useful in analysis, especially in the study of partial differential equations and distribution theory.

Definition:

A family of functions ${\phi_\epsilon(x)}_{\epsilon > 0}$ defined on $\mathbb{R}^n$ is a mollifier if it satisfies the following properties:

• Non-negativity: $\phi_\epsilon(x) \geq 0$ for all $x \in \mathbb{R}^n$ and all $\epsilon > 0$.
• Normalization: $\int_{\mathbb{R}^n} \phi_\epsilon(x) , dx = 1$ for all $\epsilon > 0$.
• Compact Support: The support of $\phi_\epsilon$ is contained in the ball $B(0, \epsilon) = {x \in \mathbb{R}^n : |x| \leq \epsilon}$, meaning $\phi_\epsilon(x) = 0$ for $|x| > \epsilon$.
• Smoothness: $\phi_\epsilon(x)$ is infinitely differentiable ($C^\infty$).

Construction:

A common way to construct a mollifier is to start with a smooth, non-negative function $\phi(x)$ satisfying $\int_{\mathbb{R}^n} \phi(x) , dx = 1$ and supp$(\phi) \subseteq B(0,1)$. Then, define

$\phi_\epsilon(x) = \frac{1}{\epsilon^n} \phi \left(\frac{x}{\epsilon}\right)$.

This scaling ensures that $\phi_\epsilon$ retains the properties of a mollifier, but with a support contained in $B(0, \epsilon)$.

Applications:

• Smoothing Functions: Convolution of a function $f$ with a mollifier $\phi_\epsilon$ yields a smooth function $f_\epsilon(x) = (f * \phi_\epsilon)(x) = \int_{\mathbb{R}^n} f(y) \phi_\epsilon(x-y) , dy$. The function $f_\epsilon$ is smoother than $f$, and as $\epsilon \to 0$, $f_\epsilon$ converges to $f$ in various senses (e.g., pointwise, in $L^p$ norm), depending on the properties of $f$.

• Regularization: Mollification is a key technique for proving results about functions that are not necessarily smooth. By approximating a non-smooth function with a sequence of smooth functions (obtained via mollification), one can often extend results proven for smooth functions to more general settings.

• Distribution Theory: Mollifiers are used to define operations on distributions, such as derivatives, and to study their properties.

• Partial Differential Equations: Mollifiers are employed in the study of weak solutions to partial differential equations, and to prove regularity results.

Key Properties and Results:

• If $f \in L^p(\mathbb{R}^n)$ for $1 \leq p < \infty$, then $f * \phi_\epsilon \to f$ in $L^p(\mathbb{R}^n)$ as $\epsilon \to 0$.

• If $f$ is continuous, then $f * \phi_\epsilon$ converges to $f$ uniformly on compact sets as $\epsilon \to 0$.

In summary, mollifiers provide a powerful tool for smoothing functions and approximating non-smooth functions with smooth ones, allowing the extension of results from smooth function spaces to more general settings.