Non-harmonic cones are Heisenberg uniqueness pairs for the Fourier transform on ${\mathbb{R}}^n$

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.

13 pages

References in corpus (1)

• Benedick's theorem for the Heisenberg group

Cited by in corpus (5)

• The Klein-Gordon equation, the Hilbert transform, and dynamics of Gauss-type maps
• Heisenberg uniqueness pairs for the Fourier transform on the Heisenberg group
• Uniqueness of the group Fourier transform on certain nilpotent Lie groups
• Heisenberg uniqueness pairs for some algebraic curves and surfaces
• Fourier nonuniqueness sets for the hyperbola and the Perron-Frobenius operators