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Polyhomogeneous mapping properties of the Radon transform and backprojection operator on the unit ball

Seiji Hansen

Abstract

This article covers polyhomogeneous mapping properties of the Radon transform $R$ of smooth functions on the open unit ball $Ω\subset\mathbb{R}^n$ and the back-projection operator $R^*$ on $Z=(-1,1)\times S^{n-1}\subset\mathbb{R}\times S^{n-1}$. We construct a double $b$-fibration which desingularizes the point-hyperplane relation of $\overlineΩ$ as the total space of a fibration over $\overline{Z}$. We provide formulas for $R$ and $R^*$ in operations generated by the associated $b$-fibrations and sharper estimates on the polyhomogeneous mapping properties of $R$ and $R^*$ compared to classic estimates using classic Mellin functional techniques. We include a discussion of a one (complex) parameter family of normal operators associated to $R$ mapping $C^{\infty}(\overlineΩ)$ to itself.

Polyhomogeneous mapping properties of the Radon transform and backprojection operator on the unit ball

Abstract

This article covers polyhomogeneous mapping properties of the Radon transform of smooth functions on the open unit ball and the back-projection operator on . We construct a double -fibration which desingularizes the point-hyperplane relation of as the total space of a fibration over . We provide formulas for and in operations generated by the associated -fibrations and sharper estimates on the polyhomogeneous mapping properties of and compared to classic estimates using classic Mellin functional techniques. We include a discussion of a one (complex) parameter family of normal operators associated to mapping to itself.
Paper Structure (14 sections, 21 theorems, 117 equations, 3 figures)

This paper contains 14 sections, 21 theorems, 117 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Statement of main results and corollaries
  3. Discussion of Theorem \ref{['MainResult']}
  4. Corollaries for normal operators constructed from $R$ and $R^*$
  5. Development of the main results
  6. Geometry of the desingularization $G$
  7. Proof of Theorem \ref{['RadonExpansion']}
  8. Proof of Theorem \ref{['MainResult']}
  9. Appendix: Background Material
  10. Manifolds with Corners
  11. Maps between Manifolds with Corners
  12. Conormal and Polyhomogeneous Function Spaces
  13. The Pullback and Pushforward Theorems
  14. Mellin Representation of a phg $b$-density

Key Result

Theorem 1

Let nonzero $a\in C^{\infty}(\partial\Omega)$ and $\tilde{a}(x) = a(\frac{x}{|x|})$ be its radial extension, $(\gamma,\ell)\in\mathbb{C}\times\mathbb{N}_0$ with $\mathop{\mathrm{Re}}\nolimits(\gamma) > -1$ and $\chi\in C^{\infty}_c[0,1)$ a cutoff $\chi\equiv1$ near $0$. For every $p\in\mathbb{N}_0$ where $\sim$ is asymptotic equivalence in the sense of asymptoticequivalence and for all $(m,k)\in\

Figures (3)

  • Figure 1: Schematic of $\overline{\Omega}$. Decomposition of $\overline{\Omega}$ along $\theta$-axis, like a topological suspension.
  • Figure :
  • Figure :

Theorems & Definitions (43)

  • Theorem 1: Phg mapping properties of $R$
  • Theorem 2: Phg mapping properties of $R^*$
  • Remark 1
  • Corollary 2.1
  • proof
  • Corollary 2.2
  • Remark 2
  • proof
  • Remark 3
  • Corollary 2.3
  • ...and 33 more