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Journal of Fourier Analysis and Applications

, Volume 24, Issue 6, pp 1425–1437 | Cite as

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

  • R. K. SrivastavaEmail author
Article
  • 169 Downloads

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.

Keywords

Bessel function Fourier transform Spherical harmonics 

Mathematics Subject Classification

Primary 42A38 Secondary 44A35 

Notes

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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Copyright information

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Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of TechnologyGuwahatiIndia

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