Bounded domains and the zero sets of Fourier transforms. (English) Zbl 0840.43012

Gindikin, Simon (ed.) et al., 75 years of Radon transform. Proceedings of the conference held at the Erwin Schrödinger International Institute for Mathematical Physics in Vienna, Austria, August 31-September 4, 1992. Cambridge, MA: International Press. Conf. Proc. Lect. Notes Math. Phys. 4, 223-239 (1994).
The purpose of this note is to study the relations between the geometry of a given domain in \(\Omega\) and the zero set \({\mathcal N} (\Omega)\) of the Fourier transform \( {\mathcal F}_{\chi_\Omega}\). The main results are: (1) description of \({\mathcal N} (\Omega)\) in terms of \(\Omega\); (2) characterization of convexity of \(\Omega\) by means of \({\mathcal N} (\Omega)\); (3) perturbation of \(\Omega\) and \({\mathcal N} (\Omega)\); (4) asymptotic behavior of the zeros of a certain class of entire functions.
For the entire collection see [Zbl 0814.00012].
Reviewer: Su Weiyi (Nanjing)


43A46 Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type