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 more precisely also with respect to its Fourier transform at each point.

Introduction

In more familiar terms, WF("f") tells not only "where" the function "f" is not "nice" (which is already described by its singular support), but also "how" or "why" it is not nice, by being more exact about the 'direction'. This concept is mostly useful in dimension at least two, therefore. The term "wave front" was coined by Lars Hörmander around 1970.

The wave front set is a closed conical subset of the cotangent bundle

:$T^*(X)$

of the differentiable manifold "X" on which the generalized function is considered.It is defined such that its projection on "X" is equal to the function's singular support w.r.t. the considered regularity (i.e. sub-presheaf of "smoother" functions).

Definition

Using local coordinates $x,xi$, the wave front set WF("f")of a generalized function "f" can be defined in the following general way:

: $\{\; m\; WF\}(f)\; =\; \{\; (x,xi)in\; T^*(X)\; mid\; xiinSigma\_x(f)\; \}$

where $Sigma\_x(f)$ is the "singular fibre of "f" above "x",which is the complement of all directions $xi$ such that the Fourier transform of "f", "localized" at "x", is sufficiently "nice" when restricted to a conical neighbourhood of $xi$.

"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. Citation unnecessary. Fact|date=February 2007)

More concretely, this can be expressed as

: $xi\; otinSigma\_x(f)\; iff\; existsphiinmathcal\; D\_x,\; exists\; Vinmathcal\; V\_xi:\; widehat\{phi\; f\}|\_Vin\; O(V)$ (or $xi=o$, never in $Sigma\_x(f)$)where
*$mathcal\; D\_x$ are compactly supported smooth functions not vanishing at "x",
*$mathcal\; V\_xi$ are "conical neighbourhoods" of $xi$, i.e. neighbourhoods "V" such that $ccdot\; Vsubset\; V$ for all $c\; >\; 0$,
*$widehat\; u|\_V$ denotes the Fourier transform of the (compactly supported generalized) function "u", restricted to "V",
*and finally $O:\; Omega\; o\; O(Omega)$ 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 $(1+|xi|)^s\; v(xi)$ belong to some L^p space.This definition makes sense, because the Fourier transform becomes moreregular (in terms of growth at infinity) when "f" is truncated with the smooth cutoff $phi$.

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 $C^infty$ 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.

Applications

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

ee also

* FBI transform
* Singular spectrum
* Essential support

References

* Lars Hörmander, "Fourier integral operators I", Acta Math. 127 (1971), pp. 79-­183.