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Continuous Parabolic Molecules

by P. Grohs and Z. Kereta

(Report number 2015-17)

Abstract
Decomposition systems based on parabolic scaling have in the last years garnered attention for their ability to answer questions regarding curvilinear singularities of functions. Well known examples of these systems are curvelets and shearlets. In recent years there has been a sufficient body of evidence to suggest that these systems are able to answer the same fundamental questions and it should thus be possible to consider them as parts of a broader framework. Thus far each such system required proofs of their properties that are tailored to their specific constructions, which is a predicament that can be avoided by focusing on the fundamental features they share. Another incentive is that while these systems exhibit same or similar properties, the specifics of their constructions might make a difference. For example, some systems are good for theoretical considerations whereas other systems might be better suited for implementations. In this paper we will construct a framework for parabolic molecules in the continuous setting, and show that it is wide enough to contain both the curvelet- and shearlet-type systems. Using almost-orthogonality we will show that some results of note (resolution of the wavefront set, microlocal Sobolev regularity) are universal for all suitable continuous parabolic molecules. The main tool we will use is that molecules are \emph{almost-orthogonal} in a certain sense.

Keywords: Curvelets, Microlocal Analysis, Shearlets, Parabolic Scaling

@Techreport{GK15_607,
  author = {P. Grohs and Z. Kereta},
  title = {Continuous Parabolic Molecules},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2015-17},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2015/2015-17.pdf },
  year = {2015}
}

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