We show that a continuous operator behaving under rotations, positive dilations, and translations like the Fourier or the Radon transform on R 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 6D(R) and ^'(R). On the other hand, a counterexample shows that the Radon transform is not determined by its behaviour above as an operator from ^(R) to ^'(S-' X R). But we can characterize the Radon trans- form as an operator acting between ^(R) and ^^(S-1 X R), the space of integrable distributions on S~' X R. In the special case n=\, our methods sharpen results of J. L. B. Cooper and H. Kober, who characterize the Fourier transform as an operator from LP(R) into LP'(R), 1 =s p *z 2. Introduction. Several authors have been concerned with the question whether an operator on function spaces on R is determined up to a constant multiple by its behaviour under rotations, dilations, and translations. For instance, Stein (15) showed that the (translation invariant) Hubert and Riesz transforms on L2 can be characterized in such a way. The Fourier and Radon transforms on R also have a well-known simple be- haviour under rotations, positive dilations, and translations. In this article, we prove that such a behaviour is characteristic: An operator acting under those operations like the Fourier or the Radon transform is a constant multiple of one of these transforms. In the case of the Fourier transform for n = 1, our methods (e.g. determining homogeneous distributions by Euler's equation) sharpen results of Cooper (1) and Kober (10), who characterized by different methods the Fourier transform as an operator from LP(R) into LP'(R), 1 <p < 2. We also obtain connections to Hardy (7) and Plancherel (12), who looked for operators on L2(R) behaving under dilations like the Fourier transform. Further, our characterizations can be used to obtain corresponding ones for operators related to the Fourier and Radon transforms (our result implies the result on Hilbert and Riesz transforms mentioned above, Stein (15)). We also remark that such characteristic properties can be applied to verify inversion formulas for the corresponding operator (in the case of the Radon