nLab
wavefront set

Idea

This page is about the concept of a wavefront set of generalized functions like hyperfunctions in the context of microlocal analysis. The wavefront set is used to further classify the kind of singularity that a generalized function exhibits in a point.

Definition

Motivation

The definition of wavefront sets is motivated by a version of a Paley-Wiener theorem that characterizes smooth compactly supported functions (${ℝ}^n\to ℝ$) by a growth condition on their Fourier transform $ℱ$:

Theorem

(Paley-Wiener for $C_0^{\mathrm{\infty }}$)

The vector space of smooth compactly supported functions is (algebraically and topologically) isomorphic, via the Fourier transform, to the space of entire functions $F$ which satisfy the following estimate: there is a positive constant $B$ such that for every integer $n>0$ there is a constant $C_n$ such that:

$$F(z)\le C_n(1+\mid z\mid {)}^{-n}\exp (B\mid Im(z)\mid )$$F(z) \le C_n (1 + |z|)^{-n} \exp{ (B \; |\operatorname{Im}(z)|)}

Smoothness

We call a smooth compactly supported function that is identical $1$ in a neighbourhood of a point $x_0$ a cutoff function at $x_0$. Let $U\subset ℝ$ be open, we identify the cotangent bundle of $U$ with $U\times {ℝ}^n$. A subset of $U\times {ℝ}^n$ is said to be conic if it stable under the transformation

$$(x,\zeta )\mapsto (x,\rho \zeta )\text{with}\rho >0$$(x, \zeta) \mapsto (x, \rho \zeta) \quad \text{with} \; \rho \gt 0

Definition

Let $f$ be a distribution and $(x_0,{\zeta }_0)$ with ${\zeta }_0\ne 0$ be a point of the cotangent bundle of $U$. f is smooth in $(x_0,{\zeta }_0)$ if there is a cutoff function $\chi$ in $x_0$ and an open cone ${\Gamma }_0$ in ${ℝ}^n$ containing ${\zeta }_0$ such that for every $m>0$ there is a nonnegative constant $C_m$ such that for all $\zeta \in {\Gamma }_0$:

$$\parallel ℱ(\chi f)(\zeta )\parallel \le C_m(1+\parallel \zeta \parallel {)}^{-m}$$\| \mathcal{F}(\chi f) (\zeta) \| \le C_m (1 + \| \zeta \|)^{-m}

where $ℱ(\chi f)$ is the Fourier transform (of the variable $\zeta$) of the function $\chi f$ (of the variable $x$).

Definition

A distribution $f$ is smooth in a conic subset $\Gamma$ of the cotangent bundle of $U$ if $f$ is smooth in a neighbourhood of every point in $\Gamma$.

Wavefront set

Let $U\subseteq {ℝ}^n$ be an open subset, $T^{*}U$ its cotangent bundle and $f$ be a distribution on $U$. The complement of the union of all conic subsets of $T^{*}U$ where $f$ is smooth is the wavefront set $\mathrm{WF}(f)$.

Examples

Onedimensional Examples

We take a brief look at distributions on $𝒟ℝ$ with singular support consisting of the origin. In one dimension, at the origin, there are of course exactly two directions along which a distribution could be smooth, namely $\zeta \le 0$ and $\zeta >0$.

We define the Fourier transform to be

$$ℱ(f)(\zeta )=\hat{f}(\zeta )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)\exp (-2\pi ix\zeta )$$\mathcal{F}(f)(\zeta) = \hat f (\zeta) = \int_{- \infty}^{\infty} f(x) \exp(-2 \pi i x \zeta)

For the delta distribution $\delta$, we have $ℱ(\delta )=1$, which does not satisfy the decay condition of smoothness.

References

• Wikipedia: wavefront set