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 | References | Similar Articles | Additional Information
Abstract: We show that a continuous operator behaving under rotations, positive dilations, and translations like the Fourier or the Radon transform on 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
and
. On the other hand, a counterexample shows that the Radon transform is not determined by its behaviour above as an operator from
to
. But we can characterize the Radon transform as an operator acting between
and
, the space of integrable distributions on
. In the special case
, our methods sharpen results of J. L. B. Cooper and H. Kober, who characterize the Fourier transform as an operator from
into
.
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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