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Transactions of the American Mathematical Society

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A characterization of Fourier and Radon transforms on Euclidean space


Author: Alexander Hertle
Journal: Trans. Amer. Math. Soc. 273 (1982), 595-607
MSC: Primary 42B10; Secondary 44A15
DOI: https://doi.org/10.1090/S0002-9947-1982-0667162-6
MathSciNet review: 667162
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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$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0667162-6
Keywords: Fourier transform, Radon transform, functional equations
Article copyright: © Copyright 1982 American Mathematical Society