Abstract
We obtain an exact formula for the Fourier transform of multiradial functions, i.e., functions of the form \(\varPhi (x)=\phi (|x_1|, \dots , |x_m|), x_i\in \mathbf R^{n_i}\), in terms of the Fourier transform of the function \(\phi \) on \(\mathbf R^{r_1}\times \cdots \times \mathbf R^{r_m}\), where \(r_i\) is either \(1\) or \(2\).
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Notes
The identity is only stated for \(\mu >-1/2\) but it is also valid for \(\mu >-3/2\) by analytic continuation.
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Acknowledgments
The first author is partially supported by supported by the ANR under the project AFoMEN no. 2011-JS01-001-01. The second author was supported by grant DMS 0900946 of the National Science Foundation of the USA. The third author was supported by the National Natural Science Foundation of China (Grant No. 11226108 and No. 11171306).
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Communicated by A. Constantin.
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Bernicot, F., Grafakos, L. & Zhang, Y. The Fourier transform of multiradial functions. Monatsh Math 175, 43–64 (2014). https://doi.org/10.1007/s00605-013-0565-3
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DOI: https://doi.org/10.1007/s00605-013-0565-3