Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Radon Transform over Tensor Fields: Injectivity, Range, and Unique Continuation Principle

Rohit Kumar Mishra, Chandni Thakkar

Abstract

A central objective in inverse problems arising in integral geometry is to understand the kernel characterization, inversion formulas, stability estimates, range characterization, and unique continuation properties of integral transforms. In this paper, we study all these aspects for Radon transforms acting on symmetric $m$-tensor fields in $\mathbb{R}^n$. Our results show that these transforms admit a coherent analytic structure, extending several key features of the classical Radon transform and tensor ray transforms to a broader geometric setting.

Radon Transform over Tensor Fields: Injectivity, Range, and Unique Continuation Principle

Abstract

A central objective in inverse problems arising in integral geometry is to understand the kernel characterization, inversion formulas, stability estimates, range characterization, and unique continuation properties of integral transforms. In this paper, we study all these aspects for Radon transforms acting on symmetric -tensor fields in . Our results show that these transforms admit a coherent analytic structure, extending several key features of the classical Radon transform and tensor ray transforms to a broader geometric setting.

Paper Structure

This paper contains 6 sections, 11 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Definitions and notations
  3. Kernel characterization and inversion of GRT
  4. Stability and Range
  5. Unique Continuation Properties
  6. Acknowledgements

Key Result

Theorem 1

[Decomposition Result, Generalised_Radon_inversion] For any tensor field $f \in H^s_t(S^m \mathbb{R}^n) (s \in \mathbb{R}, t > -n/2, m \geq 0)$, there exist uniquely defined $v_0, \dots , v_m$ with $v_i \in H^{s + i}_{t + i}(S^{m-i} \mathbb{R}^n)$ for $i = 0,1, \dots, m$ such that The estimate holds for a constant $C$ independent of $f$. The tensor field $v_0$ is known as the solenoidal part of

Theorems & Definitions (23)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Remark 1
  • Theorem 2
  • Lemma 1
  • proof
  • proof : Proof of Theorem \ref{['thm: Kernel Characterization']}
  • Lemma 2
  • proof
  • ...and 13 more