Wave front set
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.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 wheresuch that the Fourier transform of f, localized at x, is sufficiently regular when restricted to an open cone containingif 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 coordinateson the cotangent bundle, the wave front set WF(f) of a distribution ƒ can be defined in the following general way: where the singular fibresuch that the Fourier transform of f, localized at x, is sufficiently regular when restricted to a conical neighbourhood ofMore 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 thatThis definition makes sense, because the Fourier transform becomes more regular (in terms of growth at infinity) when f is truncated with the smooth cutoffThe wave front set is useful, among others, when studying propagation of singularities by pseudodifferential operators.