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


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.


Bessel function Fourier transform Spherical harmonics 

Mathematics Subject Classification

Primary 42A38 Secondary 44A35 



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.


  1. 1.
    Agranovsky, M.L., Narayanan, E.K.: Injectivity of the spherical mean operator on the conical manifolds of spheres. Siberian Math. J. 45(4), 597–605 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Agranovsky, M.L., Quinto, E.T.: Injectivity sets for the Radon transform over circles and complete systems of radial functions. J. Funct. Anal. 139(2), 383–414 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Agranovsky, M.L., Rawat, R.: Injectivity sets for spherical means on the Heisenberg group. J. Fourier Anal. Appl. 5(4), 363–372 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Agranovsky, M.L., Berenstein, C., Kuchment, P.: Approximation by spherical waves in \(L^p\)-spaces. J. Geom. Anal. 6(3), 365–383 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Agranovsky, M.L., Volchkov, V.V., Zalcman, L.A.: Conical uniqueness sets for the spherical Radon transform. Bull. Lond. Math. Soc. 31(2), 231–236 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ambartsoumian, G., Kuchment, P.: On the injectivity of the circular Radon transform. Inverse Probl. 21(2), 473–485 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)CrossRefzbMATHGoogle Scholar
  8. 8.
    Armitage, D.H.: Cones on which entire harmonic functions can vanish. Proc. R. Irish Acad. Sect. A 92(1), 107–110 (1992)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Babot, D.B.: Heisenberg uniqueness pairs in the plane, three parallel lines. Proc. Am. Math. Soc. 141(11), 3899–3904 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bagchi, S.C., Sitaram, A.: Determining sets for measures on \({\mathbb{R}}^n,\) Illinois. J. Math. 26(3), 419–422 (1982)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Benedicks, M.: On Fourier transforms of functions supported on sets of finite Lebesgue measure. J. Math. Anal. Appl. 106(1), 180–183 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Giri, D.K., Srivastava, R.K.: Heisenberg uniqueness pairs for some algebraic curves in the plane. Adv. Math. 310, 993–1016 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gröchenig, K., Jaming, P.: The Cramér-Wold theorem on quadratic surfaces and Heisenberg uniqueness pairs. arXiv:1608.06738
  14. 14.
    Hedenmalm, H., Montes-Rodríguez, A.: Heisenberg uniqueness pairs and the Klein-Gordon equation. Ann. Math. (2) 173(3), 1507–1527 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Jaming, P., Kellay, K.: A dynamical system approach to Heisenberg uniqueness pairs. arXiv:1312.6236 (2014)
  16. 16.
    Lev, N.: Uniqueness theorem for Fourier transform. Bull. Sci. Math. 135, 134–140 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Narayanan, E.K., Ratnakumar, P.K.: Benedick’s theorem for the Heisenberg group. Proc. Am. Math. Soc. 138(6), 2135–2140 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Narayanan, E.K., Thangavelu, S.: Injectivity sets for spherical means on the Heisenberg group. J. Math. Anal. Appl. 263(2), 565–579 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Narayanan, E.K., Rawat, R., Ray, S.K.: Approximation by \(K\)-finite functions in \(L^p\) spaces. Israel J. Math. 161, 187–207 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Pati, V., Sitaram, A.: Some questions on integral geometry on Riemannian manifolds, Ergodic theory and harmonic analysis (Mumbai, 1999). Sankhya Ser. A 62(3), 419–424 (2000)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Price, F.J., Sitaram, A.: Functions and their Fourier transforms with supports of finite measure for certain locally compact groups. J. Funct. Anal. 79(1), 166–182 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Sitaram, A., Sundari, M., Thangavelu, S.: Uncertainty principles on certain Lie groups. Proc. Indian Acad. Sci. Math. Sci. 105(2), 135–151 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Sjölin, P.: Heisenberg uniqueness pairs and a theorem of Beurling and Malliavin. Bull. Sci. Math. 135, 125–133 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sjölin, P.: Heisenberg uniqueness pairs for the parabola. J. Four. Anal. Appl. 19, 410–416 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Sogge, C.D.: Oscillatory integrals and spherical harmonics. Duke Math. J. 53, 43–65 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Srivastava, R.K.: Sets of injectivity for weighted twisted spherical means and support theorems. J. Fourier Anal. Appl. 18(3), 592–608 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Srivastava, R.K.: Coxeter system of lines and planes are sets of injectivity for the twisted spherical means. J. Funct. Anal. 267, 352–383 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Srivastava, R.K.: Real analytic expansion of spectral projection and extension of Hecke–Bochner identity. Israel J. Math. 200, 1–22 (2014)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32. Princeton University Press, Princeton (1971)Google Scholar
  30. 30.
    Sugiura, M.: Unitary representations and harmonic analysis, North-Holland Mathematical Library, 44. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo (1990)Google Scholar
  31. 31.
    Thangavelu, S.: An introduction to the uncertainty principle, Prog. Math. 217. Birkhauser, Boston (2004)Google Scholar
  32. 32.
    Vieli, F.J.G.: A uniqueness result for the Fourier transform of measures on the sphere. Bull. Aust. Math. Soc. 86, 78–82 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Vieli, F.J.G.: A uniqueness result for the Fourier transform of measures on the paraboloid. Matematicki Vesnik 67(1), 52–55 (2015)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of TechnologyGuwahatiIndia

Personalised recommendations