Resolution of the Wavefront Set using Continuous Shearlets
| dc.creator | Kutyniok, Gitta | |
| dc.creator | Labate, Demetrio | |
| dc.date | 2006-05-15 | |
| dc.date.accessioned | 2026-07-07T07:14:13Z | |
| dc.date.available | 2026-07-07T07:14:13Z | |
| dc.description | It is known that the continuous wavelet transform of a function $f$ decays very rapidly near the points where $f$ is smooth, while it decays slowly near the irregular points. This property allows one to precisely identify the singular support of $f$. However, the continuous wavelet transform is unable to provide additional information about the geometry of the singular points. In this paper, we introduce a new transform for functions and distributions on $\R^2$, called the Continuous Shearlet Transform. This is defined by $\mathcal{S}\mathcal{H}_f(a,s,t) = \ip{f}{ψ_{ast}}$, where the analyzing elements $ψ_{ast}$ are dilated and translated copies of a single generating function $ψ$ and, thus, they form an affine system. The resulting continuous shearlets $ψ_{ast}$ are smooth functions at continuous scales $a >0$, locations $t \in \R^2$ and oriented along lines of slope $s \in \R$ in the frequency domain. The Continuous Shearlet Transform transform is able to identify not only the location of the singular points of a distribution $f$, but also the orientation of their distributed singularities. As a result, we can use this transform to exactly characterize the wavefront set of $f$. | |
| dc.description | 31 pages, 1 figure | |
| dc.identifier | https://arxiv.org/abs/math/0605375 | |
| dc.identifier | http://arxiv.org/abs/math/0605375 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/112783 | |
| dc.subject | Functional Analysis | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 42C15; 42C40 | |
| dc.title | Resolution of the Wavefront Set using Continuous Shearlets | |
| dc.type | text |