Uniqueness of the Fourier transform on the Euclidean spaces and certain locally compact Lie groups

Abstract

We explored the Heisenberg uniqueness pairs corresponding to the spiral, hyperbola, circle, cross, exponential curves, and surfaces. Then, we prove a characterization of the Heisenberg uniqueness pairs corresponding to four parallel lines. We observe that the size of the determining sets for X depends on the number of lines and their irregular distribution that further relates to a phenomenon of interlacing of the zero sets of certain trigonometric polynomials.