Research reports

Years: 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

Radon Transform: Dual Pairs and Irreducible Representations

by G. Alberti and F. Bartolucci and F. De Mari and E. De Vito

(Report number 2020-40)

Abstract
We illustrate the general point of view we developed in an earlier paper ([SIAM J.\ Math.\ Anal., 2019]) that can be described as a variation of Helgason's theory of dual \(G\)-homogeneous pairs \((X,\Xi)\) and which allows us to prove intertwining properties and inversion formulae of many existing Radon transforms. Here we analyze in detail one of the important aspects in the theory of dual pairs, namely the injectivity of the map label-to-manifold \(\xi\to\hat\xi\) and we prove that it is a necessary condition for the irreducibility of the quasi-regular representation of \(G\) on \(L^2(\Xi)\). We further explain how our construction applies to the classical Radon and X-ray transforms in \(\mathbb R^3\).

Keywords: homogeneous spaces, Radon transform, dual pairs, square-integrable representations, inversion formula, wavelets, shearlets

@Techreport{ABDD20_913,
  author = {G. Alberti and F. Bartolucci and F. De Mari and E. De Vito},
  title = {Radon Transform: Dual Pairs and Irreducible Representations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2020-40},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2020/2020-40.pdf },
  year = {2020}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).