set is defined such that its projection on X is equal to the singular support of the function. Definition In Euclidean space, the wave front set of a distribution ƒ is defined as where is the singular fibre of ƒ at x. The singular fibre is defined to be the complement of all directions such that the Fourier transform of f, localized at x, is sufficiently regular when restricted to an open cone containing . More precisely, a direction v is in the complement of 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 is again the complement of all directions such that the Fourier transform of f, localized at x, is sufficiently regular when restricted to a conical neighbourhood of . The problem of regularity is local, and so it can be checked in the local coordinate system, using the Fourier transform on the x variables. The required regularity estimate transforms well under diffeomorphism, and so the notion of regularity is independent of the choice of local coordinates. Generalizations The notion of a wave front set can be adapted to accommodate other notions of regularity of a function. Localized can here be expressed by saying that f is truncated by some smooth cutoff function not vanishing at x. (The localization process could be done in a more elegant fashion, using germs.) More concretely, this can be expressed as where are compactly supported smooth functions not vanishing at x, are conical neighbourhoods of , i.e. neighbourhoods V such that for all , denotes the Fourier transform of the (compactly supported generalized) function u, restricted to V, is a fixed presheaf of functions (or distributions) whose choice enforces the desired regularity of the Fourier transform. Typically, sections of O are required to satisfy some growth (or decrease) condition at infinity, e.g. such that belong to some Lp space. This definition makes sense, because the Fourier