This page is about the concept of a wavefront set of generalized functions like hyperfunctions in the context of microlocal analysis. The wavefront set is used to further classify the kind of singularity that a generalized function exhibits in a point.
(Paley-Wiener for )
The vector space of smooth compactly supported functions is (algebraically and topologically) isomorphic, via the Fourier transform, to the space of entire functions which satisfy the following estimate: there is a positive constant such that for every integer there is a constant such that:
We call a smooth compactly supported function that is identical in a neighbourhood of a point a cutoff function at . Let be open, we identify the cotangent bundle of with . A subset of is said to be conic if it stable under the transformation
Let be a distribution and with be a point of the cotangent bundle of . f is smooth in if there is a cutoff function in and an open cone in containing such that for every there is a nonnegative constant such that for all :
where is the Fourier transform (of the variable ) of the function (of the variable ).
A distribution is smooth in a conic subset of the cotangent bundle of if is smooth in a neighbourhood of every point in .
Let be an open subset, its cotangent bundle and be a distribution on . The complement of the union of all conic subsets of where is smooth is the wavefront set .
We take a brief look at distributions on with singular support consisting of the origin. In one dimension, at the origin, there are of course exactly two directions along which a distribution could be smooth, namely and .
We define the Fourier transform to be
For the delta distribution , we have , which does not satisfy the decay condition of smoothness.