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Weighted Radon Transforms and Their Applications Fedor Goncharov Weighted Radon transforms and their applications Fedor Goncharov To cite this version: Fedor Goncharov. Weighted Radon transforms and their applications. General Mathematics [math.GM]. Université Paris Saclay (COmUE), 2019. English. NNT : 2019SACLX029. tel-02273044 HAL Id: tel-02273044 https://tel.archives-ouvertes.fr/tel-02273044 Submitted on 28 Aug 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. NNT : 2019SACLX029 THÈSE DE DOCTORAT de l'Université Paris-Saclay École doctorale de mathématiques Hadamard (EDMH, ED 574) Établissement d'inscription : École polytechnique Laboratoire d'accueil : Centre de mathématiques appliquées de Polytechnique, UMR 7641 CNRS Spécialité de doctorat : Mathématiques appliquées Fedor Goncharov Transformations de Radon pondérées et leurs applications Date de soutenance : 15 juillet 2019 Jan BOMAN (Stockholm University) Après avis des rapporteurs : Leonid KUNYANSKY (University of Arizona) Jury de soutenance : François ALOUGES (Professeur, École polytechnique) Président du jury Jan BOMAN (Professeur émérite, Stockholm University) Rapporteur Joonas ILMAVIRTA (Professeur adjoint, University of Jyväskylä) Examinateur François JAUBERTEAU (Professeur, Université de Nantes) Examinateur Leonid KUNYANSKY (Professeur, University of Arizona) Rapporteur Mai NGUYEN-VERGER (Professeur, Université de Cergy-Pontoise) Examinatrice Roman NOVIKOV (Professeur, École polytechnique) Directeur de thèse Alexander SHANANIN (Professeur, MIPT, MSU) Examinateur Weighted Radon transforms and their applications PhD thesis of F. O. Goncharov under the supervision of R. G. Novikov CMAP, Ecole´ Polytechnique Remerciements / Aknowledgements First of all I want to thank Roman Novikov for his very pedagogical approach in supervising this thesis. Here the word pedagogical means a lot of things, which I don't precise, but it is obvious that managing a \fresh" master student to write articles is for sure a complicated thing. I am also very thankful for our countless discussions on mathematics, philosophy and history from which I have learned not only the research field but also got a good pile of reflections on how the world works. I am also very thankful to all of the members of the jury who have agreed to come to my defense : Fran¸coisAlouges, Jan Boman, Joonas Ilmavirta, Fran¸coisJauberteau, Leonid Kunyansky, Mai Nguyen-Verger, Alexander Shananin. For me it was hard to believe that we would manage to gather such professional jury from 5 different countries ! Of course, I could write a small \passage" on each member, instead I will just say that I am glad that I had met you at some points of my life and had a chance to have discussions with you. In turn, the defense would not have happened if jury would have not been welcomed in Ecole´ polytechnique. At this point I would like to thank the administration of CMAP for the hospitality and warm atmosphere : Anne de Bouard (for recommending me for this PhD position and accepting me during the \conseils du labo" when my french was not good at all), Naar Nass´era(for being so nice, so that I always preferred always to go an extra time to your office than sending emails), Alexandra Noiret (for coping with me and my jury who do not really followed the rules \comme il faut"). Additionally, thank you Alexandra for dealing with invitations, it was not easy at all ! It was very pleasant to work in my bureau because of my friends with whom we mostly did things in order not to work ! Thank you Martin for drugging me (and the whole lab) on chess, Heythem (for keeping the bureau in order and for our discussions on literature, so that you was reading russian authors and I started to read french ones), Celine (for feeding me with food intended for the PhD seminar), Juliette and Pierre (for being my personal ma^ıtres of french language, literature and beer), Arthur (for explaining politics in France and the model of electric cars), Antoine and H´el`ene(for just being best friends in so many occasions), Kevish (for your stories about Mauritius Islands), Beno^ıt(for discussions of history and fu- ture discussions of physics), Florian (for stories about trail running and football), Younes (for our math discussions), Etienne´ and Kuankuan (for being best and most non-trivial flat neighboors ever), Alexey (for millions of hours of discussions of mathematics, literature, phi- losophy and beers near the lake), Misha Isaev (for bringing me the idea to study tomography in France). I am also thankful to all members of the laboratory with whom I crossed in the cantine, seminars, football matches and evenings in Paris, I will keep these memories for a long time ! Finally, I want to thank my family : Yegor (in particular, for turning on the projector before the defense), my mother and father (for everything in my life), Marie-Christine (for providing the best pat´efor my \p^otde th`ese"),Martine, Roger, Florent (for french-spiritual support). However, all this work would not be possible without support of my lovely wife, her help with the french part of the thesis and, of course, her perfect dinners ! Thank you Marie- Liesse, you are the best I could ever have and I hope it won't be a second thesis ! P. S. Additionally I want to thank studio \Europacorp" of Luc Besson for financing the costume I was wearing during the defense. 1 Contents 1 Introduction (en fran¸cais) 8 1 Un interm`edehistorique `ala transformation de type Radon . .8 1.1 L'h´eritagede Minkowski-Funk-Radon . .8 1.2 Red´ecouvertes, les Beatles et le prix Nobel . 10 1.3 Remarques finales . 15 2 Pr´eliminairespour les transformations de type Radon pond´er´ees . 15 3 Probl`emeset motivation . 17 4 R´esum´ede nos r´esultats . 19 4.1 R´esum´edes r´esultatsde la partie I . 19 4.2 R´esum´edes r´esultatsde la partie II . 60 5 Conclusions . 66 2 Introduction (in English) 69 1 A historical interlude to Radon-type transforms . 69 1.1 Minkowski-Funk-Radon's heritage . 69 1.2 Rediscoveries, The Beatles and the Nobel Prize Award . 71 1.3 Finalizing remarks . 75 2 Preliminaries for weighted Radon-type transforms . 75 3 Problems and motivation . 77 4 Summary of our results . 78 4.1 Summary of results from Part I . 78 4.2 Summary of results from Part II . 117 5 Conclusions . 123 I Inversions of weighted Radon transforms in 3D 132 3 Article 1. An analog of Chang inversion formula for weighted Radon transforms in multidimensions 133 1 Introduction . 133 2 Chang-type formulas in multidimensions . 134 3 Weighted Radon transforms in 3D in tomographies . 135 4 Proof of Theorem 1 . 135 4.1 Proof of sufficiency . 136 4.2 Proof of necessity . 137 5 Proof of Lemma 1 . 138 5.1 The case of odd n ............................. 138 5.2 The case of even n ............................ 139 6 Proof of Lemma 2 . 139 2 4 Article 2. An iterative inversion of weighted Radon transforms in multidimensions 142 1 Introduction . 142 2 Some preliminary results . 145 2.1 Some formulas for R and R−1 ...................... 145 2.2 Symmetrization of W ........................... 146 2.3 Operators Q and numbers σ ................. 147 Wf ;D;m Wf ;D;m 3 Main results . 149 3.1 Case of σ < 1............................ 149 Wf ;D;1 3.2 Case of 1 ≤ σ < +1 ........................ 149 Wf ;D;1 3.3 Exact inversion for finite Fourier series weights . 150 3.4 Additional comments . 151 4 Generalization to multidimensions . 152 5 Proofs of Lemma 1, 2, 3, 4 . 155 5.1 Proof of Lemma 1 . 155 5.2 Proof of Lemma 2 . 156 5.3 Proof of Lemma 3 . 157 5.4 Proof of Lemma 4 . 157 6 Proofs of Theorems 1, 2, 3 . 158 6.1 Proof of Theorem 1 . 158 6.2 Proof of Theorem 3 . 158 6.3 Proof of Theorem 2 . 158 7 Acknowledgments . 160 II A breakdown of injectivity 162 5 Article 3. A counterexample to injectivity for weighted Radon transforms163 1 Introduction . 163 2 Relations between the Radon and the ray transforms . 164 3 Boman's example . 165 4 Main results . 166 5 Proof of formula (5.20) . 167 6 Acknowledgments . 167 6 Article 4. An example of non-uniqueness for Radon transforms with pos- itive continuous rotation invariant weights 169 1 Introduction . 169 2 Some preliminaries . 171 3 Main results for d =3 .............................. 172 3.1 Construction of f ............................. 172 3.2 Construction of W ............................ 173 3.3 Construction of W0 ............................ 174 3.4 Construction of W1;:::;WN and ξ0; : : : ; ξN ............... 175 d;2 4 Extension to the case of RW ........................... 177 5 Proofs of Lemma 1 and formula (6.33) . 178 5.1 Proof of Lemma 1 . 178 5.2 Proof of formula (6.33) . 179 6 Proof of Lemma 2 . 179 7 Proof of Lemma 3 . 181 3 8 Proof of Lemma 4 . 182 8.1 Proof of estimate (6.88) . 182 8.2 Proof of estimate (6.90) . 183 8.3 Proof of estimate (6.89) . 185 7 Article 5. A breakdown of injectivity for weighted ray transforms 188 1 Introduction . 188 2 Some preliminaries . 191 3 Main results . 192 3.1 Construction of f and W for d =2 ................... 193 3.2 Construction of W and f for d ≥ 3 ................... 198 4 Proof of Theorem 2 . 198 4.1 Proof for d ≥ 3 .............................
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