Wave Front Set
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High Quality Content by WIKIPEDIA articles! 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 whose singular support is concentrated on a smoo...
High Quality Content by WIKIPEDIA articles! 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 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 to be smooth in the direction of v. Otherwise, v is in the wavefront set.
