A characterization of Fourier and Radon transforms on Euclidean space

Abstract:

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 ${L^p}^\prime ({\mathbf {R}}),1 \leqslant p \leqslant 2$.