The Euclidean Radon transform on R²

Write $C_c^{\mathrm{\infty }}({\mathbb{R}}^2;\mathbb{C})$ for the complex-valued smooth functions of compact support on the Euclidean plane. For $u\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n;\mathbb{C})$, $n=1$ or $2$, use the compact-support Riemann Fourier transform $\widehat{u}(\xi )={\int }_{{\mathbb{R}}^n}u(x)e^{-2\pi ix\cdot \xi }dx\mathrm{.}$ For $\theta =({\theta }_1,{\theta }_2)\in S^1:=\{x\in {\mathbb{R}}^2:\mathrm{\mid }x\mathrm{\mid }=1\}$ put ${\theta }^{\perp }=(-{\theta }_2,{\theta }_1)$. The line with normal $\theta$ and signed offset $s$ is $\{x\in {\mathbb{R}}^2:x\cdot \theta =s\}$. Its Euclidean Radon transform is $Rf(\theta ,s):={\int }_{\mathbb{R}}f(s\theta +t{\theta }^{\perp })dt,f\in C_c^{\mathrm{\infty }}({\mathbb{R}}^2;\mathbb{C}),(\theta ,s)\in S^1\times \mathbb{R}\mathrm{.}$ The line integral is a compact-support Riemann integral. The full data set uses every oriented normal $\theta \in S^1$ and every $s\in \mathbb{R}$; the same unoriented line is represented by $(\theta ,s)$ and $(-\theta ,-s)$. For a one-dimensional function $h\in C_c^{\mathrm{\infty }}(\mathbb{R};\mathbb{C})$, define $\mathrm{\mid }{D}_s\mathrm{\mid }h(s):={\int }_{\mathbb{R}}2\pi \mathrm{\mid }\sigma \mathrm{\mid }\widehat{h}(\sigma )e^{2\pi is\sigma }d\sigma \mathrm{.}$ For $f\in C_c^{\mathrm{\infty }}({\mathbb{R}}^2;\mathbb{C})$ define the Riesz potential notation used here by $\mathrm{\mid }D{\mathrm{\mid }}^{-1}f(x):={\int }_{{\mathbb{R}}^2}{e}^{2\pi ix\cdot \xi }\frac{\widehat{f}(\xi )}{2\pi \mathrm{\mid }\xi \mathrm{\mid }}d\xi \mathrm{.}$ The quotient in the last integral is assigned value $0$ at $\xi =0$. These integrals are absolutely convergent on the stated classes: compact-support integration by parts gives rapid decay of $\widehat{f}$ and $\widehat{h}$, while $\widehat{f}$ is bounded near $0$ and $\mathrm{\mid }\xi {\mathrm{\mid }}^{-1}$ is locally integrable in two dimensions. Here the multiplier for $\mathrm{\mid }{D}_s\mathrm{\mid }$ is $2\pi \mathrm{\mid }\sigma \mathrm{\mid }$, corresponding to the $e^{-2\pi is\sigma }$ Fourier convention.

The Fourier transform on an LCA group

Mikko Salo, RICAM lecture notes v1, §2.1

Frank Natterer, “X-ray Tomography,” §§2.1–2.2