Wave front set
Encyclopedia
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.
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
with φ(x0) ≠ 0 and an open cone Γ containing v such that the estimate
holds for all positive integers N. Here
denotes the Fourier transform. Observe that the wavefront set is a conical
set, since if (x,v) ∈ Wf(ƒ), then (x,λv) ∈ 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.
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.
cutoff function not vanishing at x. (The localization process could be done in a more elegant fashion, using germ
s.)
More concretely, this can be expressed as
where
Typically, sections of O are characterized by some growth (or decrease) condition at infinity, e.g. such that
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
.
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.
functions,
we must take for O(Ω) the classical function spaces called O′M(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
.
of singularities
by pseudodifferential operators.
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, more precisely in microlocal analysis
Microlocal analysis
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...
, the wave front (set) WF(f) characterizes the singularities
Mathematical singularity
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
of a generalized function
Generalized function
In mathematics, generalized functions are objects generalizing 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 physical phenomena such as point charges...
f, not only in space
Space
Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...
, but also with respect to its Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
at each point. The term "wave front" was coined by Lars Hörmander
Lars Hörmander
Lars Valter Hörmander is a Swedish mathematician who has been called "the foremost contributor to the modern theory of linear partial differential equations". He was awarded the Fields Medal in 1962, and the Wolf Prize in 1988...
around 1970.
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
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, the wave front set of ƒ is defined as the complement
Complement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
of the set of all pairs (x0,v) such that there exists a test function
with φ(x0) ≠ 0 and an open cone Γ containing v such that the estimateholds for all positive integers N. Here
denotes the Fourier transform. Observe that the wavefront set is a conicalCone (topology)
In topology, especially algebraic topology, the cone CX of a topological space X is the quotient space:CX = /\,of the product of X with the unit interval I = [0, 1]....
set, since if (x,v) ∈ Wf(ƒ), then (x,λv) ∈ 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
Differentiable manifold
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...
X. In this more general situation, the wave front set is a closed conical subset of the cotangent bundle
Cotangent bundle
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...
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.
Definition
In Euclidean space, the wave front set of a distributionDistribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. 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...
ƒ is defined as
where
is the singular fibre of ƒ at x. The singular fibre is defined to be the complementComplement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
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
Differentiable manifold
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...
M, using local coordinates
on the cotangent bundleCotangent bundle
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...
, 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 diffeomorphismDiffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
, 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 smoothSmooth
Smooth means having a texture that lacks friction. Not rough.Smooth may also refer to:-In mathematics:* Smooth function, a function that is infinitely differentiable; used in calculus and topology...
cutoff function not vanishing at x. (The localization process could be done in a more elegant fashion, using germ
Germ (mathematics)
In mathematics, the notion of a germ of an object in/on a topological space captures the local properties of the object. In particular, the objects in question are mostly functions and subsets...
s.)
More concretely, this can be expressed as
-
(or 
, never in 
)
where
are compactly supported smooth functionSmooth functionIn mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
s not vanishing at x,
are conical neighbourhoods of 
, i.e. neighbourhoodNeighbourhood (mathematics)In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can move that point some amount without leaving the set.This concept is closely related to the...
s V such that
for all 
,
denotes the Fourier transformFourier transformIn mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
of the (compactly supported generalized) function u, restricted to V,- and finally
is the presheaf characterizing the regularity of the Fourier transform.
Typically, sections of O are characterized by some growth (or decrease) condition at infinity, e.g. such that
belong to some Lp spaceLp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
.
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 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 functions,we must take for O(Ω) the classical function spaces called O′M(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
Smooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
.
Applications
The wave front set is useful, among others, when studying propagationWave propagation
Wave propagation is any of the ways in which waves travel.With respect to the direction of the oscillation relative to the propagation direction, we can distinguish between longitudinal wave and transverse waves....
of singularities
Mathematical singularity
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
by pseudodifferential operators.