In mathematical analysis, more precisely in microlocal analysis, the wave front (set) WF(f) characterizes the singularities of a generalized function f, not only in space, but also with respect to its Fourier transform at each point.
The complementary notion of a function being non-singular in a direction is microlocal smoothness.
(x0) ≠ 0 and an open cone Γ containing v such that the estimate holds for all positive integers N. Here
In the example discussed in the previous paragraph, the wavefront set is the set-theoretic complement of the image of the tangent bundle of the curve inside the tangent bundle of the plane.
Because the definition involves cutoff by a compactly supported function, the notion of a wave front set can be transported to any differentiable manifold X.
In this more general situation, the wave front set is a closed conical subset of the cotangent bundle T*(X), since the ξ variable naturally localizes to a covector rather than a vector.
The wave front set is defined such that its projection on X is equal to the singular support of the function.
In Euclidean space, the wave front set of a distribution ƒ is defined as where
such that the Fourier transform of f, localized at x, is sufficiently regular when restricted to an open cone containing
if there is a compactly supported smooth function φ with φ(x) ≠ 0 and an open cone Γ containing v such that the following estimate holds for each positive integer N: Once such an estimate holds for a particular cutoff function φ at x, it also holds for all cutoff functions with smaller support, possibly for a different open cone containing v. On a differentiable manifold M, using local coordinates
on the cotangent bundle, the wave front set WF(f) of a distribution ƒ can be defined in the following general way: where the singular fibre
such that the Fourier transform of f, localized at x, is sufficiently regular when restricted to a conical neighbourhood of
Localized can here be expressed by saying that f is truncated by some smooth cutoff function not vanishing at x.
More concretely, this can be expressed as where Typically, sections of O are required to satisfy some growth (or decrease) condition at infinity, e.g. such that
This definition makes sense, because the Fourier transform becomes more regular (in terms of growth at infinity) when f is truncated with the smooth cutoff
The wave front set is useful, among others, when studying propagation of singularities by pseudodifferential operators.
The propagation of singularities theorem characterizes the wave front set.