cutoff function vs mollifiers

Question

$\boldsymbol{Q_1}$ What are cutoff functions? What are mollifiers? I cannot distinguish the two. Could anyone give some concrete/simple examples of cutoff functions and how they differ from mollifiers? $\boldsymbol{\text{I did check wiki (so please, I do not want wiki type answer)}}$

$\boldsymbol{Q_2}$ How to obtain compact support using a cutoff function? Could anyone give a example? Or take a look at the last sentence of Lemma 1.5 and explain what it means mathematically?

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Many Thanks!

Answer 1

A mollifier is a function $f$ that you convolve with another function $g$ to get a function which is "close" to $g$ but "nicer". For instance $g$ might be a general $L^1$ function and $g*f$ might be a smooth, compactly supported approximation to $g$. Really a mollifier is not one function but a sequence, or even sometimes a one-parameter continuous family.

A cutoff function is a function which is usually smooth, $1$ on some set $K$ of interest, and $0$ outside a slightly larger set $A$ of interest. You multiply it with a function which is usually smooth but has support outside of $A$ to get an approximation which is supported only inside $A$ but equal to the original function on $K$. Cutoff functions are commonly constructed as the mollification of an indicator function.

Answer 2

There are a few related concepts here:

1) A bump function is a term for a smooth function with compact support. The set of all bump functions forms a vector space. If these functions are on $\mathbb{R}^n$, then it is often denoted $C_c^\infty(\mathbb{R}^n)$. In distribution theory, this is what is most commonly referred to when one refers to test functions, but this second usage depends on context.

2) A smooth cutoff function is a bump function, but with additional properties. Namely, a smooth cutoff function $f$ should satisfy $0 \leq f \leq 1$ everywhere, equal to $1$ on a small neighborhood of a compact set $K$, and equal to $0$ outside of a compact set $K'$. A general cutoff function has these properties but is only required to be continuous. The existence of cutoff functions is the subject of a famous theorem, Urysohn's lemma.

3) A mollifier is a function $\phi$ satisfying certain properties: $\phi$ is smooth, compactly supported (i.e. a bump function), integrates to $1$, and satisfies $$ \lim_{\varepsilon\to 0} \frac{1}{\varepsilon^n}\phi(x/\varepsilon) = \delta(x) $$ where $\delta$ denotes the Dirac delta and the equation is interpreted in the sense of distributions; namely, it should hold when integrated against a suitable sense of test functions, such as $C_c^\infty(\mathbb{R}^n)$ or the Schwarz functions. This is the definition taken in the Wikipedia article on mollifiers. The function given in the section "Concrete example" is sometimes called the standard mollifier.

4) An approximate identity is pretty much a mollifier, but we don't require it to be compactly supported. Also, an approximate identity can be defined as a single function, or as a sequence of functions. For example, the Poisson kernel on the half-space, $$ \mathcal{P}(x) = \frac{c_n}{(1+|x|^2)^{\frac{n+1}{2}}} $$ (where $c_n$ normalizes the function so it integrates to $1$) is an approximate identity that is defined using a single function; one then obtains a sequence of approximations to the identity by defining $$ \mathcal{P}_y(x) = \frac{1}{y^n}\mathcal{P}(x/y) = \frac{c_ny}{(y^2 + |x|^2)^{\frac{n+1}{2}}}. $$ (This function can also be what is meant when one hears "Poisson kernel.") Notice how $\mathcal{P}$ and $\mathcal{P}_y$ do not have compact support. Yet it enjoys most of the properties a compactly supported mollifier would have: in particular, the "smoothing property" and the "approximation of identity" properties are satisfied (with appropriate changes made). Other good examples of approximate identities are the Fejer kernel and the heat kernel, AKA the Gauss-Weierstrass kernel.

There is some blending of vocabulary between 3) and 4). The way we have defined things, every mollifier is an approximate identity, but not conversely. Sometimes when someone says "approximate identity" they mean "mollifier" in the sense described here. But I think it's comparatively rare to hear the term "mollifier" for a kernel without compact support. Approximate identity also has a definition in functional analysis, which should generalize some of these notions in the $C^*$-algebra setting. The point is, whether you distinguish 3) and 4) is pretty much up to personal preference.

These concepts are also related to each other via convolutions. A common way to construct a smooth cutoff function is to take the convolution of a characteristic function (AKA indicator function) with a mollifier or an approximate identity, and use the fact that this convolution approximates the original function pointwise under suitable assumptions.