Heisenberg uniqueness pairs for some algebraic curves in the plane

Abstract

A Heisenberg uniqueness pair is a pair $\left(\Gamma ,\Lambda \right)$, where $\Gamma$ is a curve and $\Lambda$ is a set in ${\mathbb{R}}^2$ such that whenever a finite Borel measure $\mu$ having support on $\Gamma$ which is absolutely continuous with respect to the arc length on $\Gamma$ satisfies $\hat{\mu }{|}_{\Lambda }=0,$ then it is identically $0.$ In this article, we investigate the Heisenberg uniqueness pairs corresponding to the spiral, hyperbola, circle and certain exponential curves. Further, we work out a characterization of the Heisenberg uniqueness pairs corresponding to four parallel lines. In the latter case, we observe a phenomenon of interlacing of three trigonometric polynomials.

23 pages. Accepted in Advances in Mathematics

References in corpus (1)

• Benedick's theorem for the Heisenberg group

Cited by in corpus (1)

• Uniqueness of the group Fourier transform on certain nilpotent Lie groups