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Ridgelet-type frame decompositions for Sobolev spaces related to linear transport

by P. Grohs

(Report number 2010-46)

Abstract
In this paper we study stability properties of ridgelet and curvelet frames for mixed-smoothness Sobolev spaces with norm kfks=kfkL2(Rd)+ks.VfkL2(Rd). Here s E Sd-1 is transport direction and V denotes the gradient of f. Such spaces arise as domains of linear, rst order transport equations. The main result of this paper is that ridgelet frames are stable in k.ks regardless of s, while curvelet frames are not. To show the second statement we explicitly construct functions f,g whose curvelet coeffcients have all the same modulus but kfks<8 and kgks=8.

Keywords: Sobolev Spaces, Ridgelets, Curvelets, Transport Equations

@Techreport{G10_444,
  author = {P. Grohs},
  title = {Ridgelet-type frame decompositions for Sobolev spaces related to linear transport },
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2010-46},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2010/2010-46.pdf },
  year = {2010}
}

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