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Non-harmonic Cones are Heisenberg Uniqueness Pairs for the Fourier Transform on \({\mathbb {R}}^n\)

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Abstract

In this article, we prove that a cone is a Heisenberg uniqueness pair corresponding to sphere as long as the cone does not completely recline on the level surface of any homogeneous harmonic polynomial on \({\mathbb {R}}^n\). We derive that \(\left( S^2, \text { paraboloid}\right) \) and \(\left( S^2, \text { geodesic of } S_r(o)\right) \) are Heisenberg uniqueness pairs for a class of certain symmetric finite Borel measures in \({\mathbb {R}}^3\). Further, we correlate the problem of Heisenberg uniqueness pairs to the sets of injectivity for the spherical mean operator.

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Acknowledgements

The author wishes to thank E. K. Narayanan and Rama Rawat for several fruitful discussions. The author would also like to gratefully acknowledge the support provided by IIT Guwahati, Government of India.

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  1. Department of Mathematics, Indian Institute of Technology, Guwahati, 781039, India

    R. K. Srivastava

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  1. R. K. Srivastava
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Correspondence to R. K. Srivastava.

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Communicated by Alex Iosevich.

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Srivastava, R.K. Non-harmonic Cones are Heisenberg Uniqueness Pairs for the Fourier Transform on \({\mathbb {R}}^n\). J Fourier Anal Appl 24, 1425–1437 (2018). https://doi.org/10.1007/s00041-018-9601-y

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  • DOI: https://doi.org/10.1007/s00041-018-9601-y

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