# Wave front set

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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 term "wave front" was coined by Lars Hörmander around 1970.

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

In mathematical analysis, microlocal analysis comprises techniques developed from the 1950s onwards based on Fourier transforms related to the study of variable-coefficients-linear and nonlinear partial differential equations. This includes generalized functions, pseudo-differential operators, wave front sets, Fourier integral operators, oscillatory integral operators, and paradifferential operators.

In mathematics, generalized functions, or distributions, are objects extending the notion of functions. There is more than one recognized theory. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and describing discrete physical phenomena such as point charges. They are applied extensively, especially in physics and engineering.

## Introduction

In more familiar terms, WF(f) tells not only where the function f is singular (which is already described by its singular support), but also how or why it is singular, by being more exact about the direction in which the singularity occurs. This concept is mostly useful in dimension at least two, since in one dimension there are only two possible directions. The complementary notion of a function being non-singular in a direction is microlocal smoothness.

Intuitively, as an example, consider a function ƒ whose singular support is concentrated on a smooth curve in the plane at which the function has a jump discontinuity. In the direction tangent to the curve, the function remains smooth. By contrast, in the direction normal to the curve, the function has a singularity. To decide on whether the function is smooth in another direction v, one can try to smooth the function out by averaging in directions perpendicular to v. If the resulting function is smooth, then we regard ƒ to be smooth in the direction of v. Otherwise, v is in the wavefront set.

Formally, in Euclidean space, the wave front set of ƒ is defined as the complement of the set of all pairs (x0,v) such that there exists a test function ${\displaystyle \phi \in C_{c}^{\infty }}$ with ${\displaystyle \phi }$(x0)  0 and an open cone Γ containing v such that the estimate

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and similar spaces of higher dimension. It is named after the Ancient Greek mathematician Euclid of Alexandria. The term "Euclidean" distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions.

In set theory, the complement of a set A refers to elements not in A.

${\displaystyle |(\phi f)^{\wedge }(\xi )|\leq C_{N}(1+|\xi |)^{-N}\quad {\mbox{for all }}\ \xi \in \Gamma }$

holds for all positive integers N. Here ${\displaystyle (\phi f)^{\wedge }}$ denotes the Fourier transform. Observe that the wavefront set is conical in the sense that if (x,v)  Wf(ƒ), then (xv)  Wf(ƒ) for all λ > 0. 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 mathematics, a differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.

In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle.

## Definition

In Euclidean space, the wave front set of a distribution ƒ is defined as

Distributions are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are distributions, such as the Dirac delta function.

${\displaystyle {\rm {WF}}(f)=\{(x,\xi )\in \mathbb {R} ^{n}\times \mathbb {R} ^{n}\mid \xi \in \Sigma _{x}(f)\}}$

where ${\displaystyle \Sigma _{x}(f)}$ is the singular fibre of ƒ at x. The singular fibre is defined to be the complement of all directions ${\displaystyle \xi }$ such that the Fourier transform of f, localized at x, is sufficiently regular when restricted to an open cone containing ${\displaystyle \xi }$. More precisely, a direction v is in the complement of ${\displaystyle \Sigma _{x}(f)}$ 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:

${\displaystyle (\phi f)^{\wedge }(\xi )

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 ${\displaystyle x,\xi }$ on the cotangent bundle, the wave front set WF(f) of a distribution ƒ can be defined in the following general way:

${\displaystyle {\rm {WF}}(f)=\{(x,\xi )\in T^{*}(X)\mid \xi \in \Sigma _{x}(f)\}}$

where the singular fibre ${\displaystyle \Sigma _{x}(f)}$ is again the complement of all directions ${\displaystyle \xi }$ such that the Fourier transform of f, localized at x, is sufficiently regular when restricted to a conical neighbourhood of ${\displaystyle \xi }$. 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

${\displaystyle \xi \notin \Sigma _{x}(f)\iff \xi =0{\text{ or }}\exists \phi \in {\mathcal {D}}_{x},\ \exists V\in {\mathcal {V}}_{\xi }:{\widehat {\phi f}}|_{V}\in O(V)}$

where

• ${\displaystyle {\mathcal {D}}_{x}}$ are compactly supported smooth functions not vanishing at x,
• ${\displaystyle {\mathcal {V}}_{\xi }}$ are conical neighbourhoods of ${\displaystyle \xi }$, i.e. neighbourhoods V such that ${\displaystyle c\cdot V\subset V}$ for all ${\displaystyle c>0}$,
• ${\displaystyle {\widehat {u}}|_{V}}$ denotes the Fourier transform of the (compactly supported generalized) function u, restricted to V,
• ${\displaystyle O:\Omega \to O(\Omega )}$ 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 ${\displaystyle (1+|\xi |)^{s}v(\xi )}$ belong to some Lp space. 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 ${\displaystyle \phi }$.

The most difficult "problem", from a theoretical point of view, is finding the adequate sheaf O characterizing functions belonging to a given subsheaf E of the space G of generalized functions.

### Example

If we take G = D the space of Schwartz distributions and want to characterize distributions which are locally ${\displaystyle C^{\infty }}$ functions, we must take for O(Ω) the classical function spaces called OM(Ω) in the literature.

Then the projection on the first component of a distribution's wave front set is nothing else than its classical singular support, i.e. the complement of the set on which its restriction would be a smooth function.

### Applications

The wave front set is useful, among others, when studying propagation of singularities by pseudodifferential operators.

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## References

• Lars Hörmander, Fourier integral operators I, Acta Math. 127 (1971), pp. 79–183.
• Hörmander, Lars (1990), The Analysis of Linear Partial Differential Equations I: Distribution Theory and Fourier Analysis, Grundlehren der mathematischen Wissenschaften, 256 (2nd ed.), Springer, pp. 251–279, ISBN   0-387-52345-6 Chapter VIII, Spectral Analysis of Singularities