In this paper we study the Radon transform $R$ on the set $\mathscr{H}$ of horocycles of a homogeneous tree $T$, and describe its image on various function spaces. We show that the functions of compact support on $\mathscr{H}$ that satisfy two explicit Radon conditions constitute the image under $R$ of functions of finite support on $T$. We extend these results to spaces of functions with suitable decay on $T$, whose image under $R$ satisfies corresponding decay conditions and contains distributions on $\mathscr{H}$ that are not defined pointwise....

We prove that ridgelet transform $R:𝒮\left({ℝ}^2\right)\to 𝒮\left(𝕐\right)$ and adjoint ridgelet transform $R^{*}:𝒮\left(𝕐\right)\to 𝒮\left({ℝ}^2\right)$ are continuous, where $𝕐={ℝ}^{+}\times ℝ\times [0,2\pi ]$. We also define the ridgelet transform $ℛ$ on the space ${𝒮}^{\text{'}}\left({ℝ}^2\right)$ of tempered distributions on ${ℝ}^2$, adjoint ridgelet transform ${ℛ}^{*}$ on ${𝒮}^{\text{'}}\left(𝕐\right)$ and establish that they are linear, continuous with respect to the weak${}^{*}$-topology, consistent with $R$, $R^{*}$ respectively, and they satisfy the identity $({ℛ}^{*}\circ ℛ)\left(u\right)=u$, $u\in {𝒮}^{\text{'}}\left({ℝ}^2\right)$.