nLab
wavefront set

Contents

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 ( n\mathbb{R}^n \to \mathbb{R}) by a growth condition on their Fourier transform \mathcal{F}:

Theorem

(Paley-Wiener for C 0 C^{\infty}_0)

The vector space of smooth compactly supported functions is (algebraically and topologically) isomorphic, via the Fourier transform, to the space of entire functions FF which satisfy the following estimate: there is a positive constant BB such that for every integer n>0n \gt 0 there is a constant C nC_n such that:

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

Smoothness

We call a smooth compactly supported function that is identical 11 in a neighbourhood of a point x 0x_0 a cutoff function at x 0x_0. Let UU \subset \mathbb{R} be open, we identify the cotangent bundle of UU with U× nU \times \mathbb{R}^n. A subset of U× nU \times \mathbb{R}^n is said to be conic if it stable under the transformation

(x,ζ)(x,ρζ)withρ>0 (x, \zeta) \mapsto (x, \rho \zeta) \quad \text{with} \; \rho \gt 0
Definition

Let ff be a distribution and (x 0,ζ 0)(x_0, \zeta_0) with ζ 00\zeta_0 \neq 0 be a point of the cotangent bundle of UU. f is smooth in (x 0,ζ 0)(x_0, \zeta_0) if there is a cutoff function χ\chi in x 0x_0 and an open cone Γ 0\Gamma_0 in n\mathbb{R}^n containing ζ 0\zeta_0 such that for every m>0m \gt 0 there is a nonnegative constant C mC_m such that for all ζΓ 0\zeta \in \Gamma_0:

(χf)(ζ)C m(1+ζ) m \| \mathcal{F}(\chi f) (\zeta) \| \le C_m (1 + \| \zeta \|)^{-m}

where (χf)\mathcal{F}(\chi f) is the Fourier transform (of the variable ζ\zeta) of the function χf\chi f (of the variable xx).

Definition

A distribution ff is smooth in a conic subset Γ\Gamma of the cotangent bundle of UU if ff is smooth in a neighbourhood of every point in Γ\Gamma.

Wavefront set

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

Examples

Onedimensional Examples

We take a brief look at distributions on 𝒟\mathcal{D} \mathbb{R} 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 ζ0\zeta \le 0 and ζ>0\zeta \gt 0.

We define the Fourier transform to be

(f)(ζ)=f^(ζ)= f(x)exp(2πixζ) \mathcal{F}(f)(\zeta) = \hat f (\zeta) = \int_{- \infty}^{\infty} f(x) \exp(-2 \pi i x \zeta)

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

References

Revised on May 3, 2010 10:16:45 by Tim van Beek (192.76.162.8)