A characterization of Fourier and Radon transforms on Euclidean space

We show that a continuous operator behaving under rotations, positive dilations, and translations like the Fourier or the Radon transform on ${{\mathbf{R}}^n}$ must be a constant multiple of one of these transforms. We prove this characterization for various function spaces, e.g. we characterize the Fourier transform as an operator acting on spaces between $\mathfrak{D}({{\mathbf{R}}^n})$ and $\mathfrak{D}'({{\mathbf{R}}^n})$. On the other hand, a counterexample shows that the Radon transform is not determined by its behaviour above as an operator from $\mathfrak{D}({{\mathbf{R}}^n})$ to $\mathfrak{D}'({S^{n - 1}} \times {\mathbf{R}})$. But we can characterize the Radon transform as an operator acting between $\mathfrak{D}({{\mathbf{R}}^n})$ and ${\mathfrak{D}'_{{L^1}}}({S^{n - 1}} \times {\mathbf{R}})$, the space of integrable distributions on ${S^{n - 1}} \times {\mathbf{R}}$. In the special case $n = 1$, our methods sharpen results of J. L. B. Cooper and H. Kober, who characterize the Fourier transform as an operator from ${L^p}({\mathbf{R}})$ into ${L^p}^\prime ({\mathbf{R}}),1 \leqslant p \leqslant 2$.