Kobayashi, Toshiyuki Asymptotic behaviour of the null variety for a convex domain in a non- positively curved space form. (English) Zbl 0701.32008 J. Fac. Sci., Univ. Tokyo, Sect. I A 36, No. 3, 389-478 (1989). The null variety of a measurable bounded set \(\Omega \subset {\mathbb{R}}^ n\) is defined as the analytic set \[ N(\Omega)=\{\xi \in {\mathbb{C}}^ n,\quad {\tilde \chi}_{\Omega}(\xi)=0\}, \] where \({\tilde \chi}{}_{\Omega}\) is the Fourier transform of the characteristic function \(\chi_{\Omega}\) of \(\Omega\). The author poses and discusses the following problems:1. Describe N(\(\Omega\)) in terms of the geometric invariants of \(\Omega\). 2. Does N(\(\Omega\)) determine \(\Omega\) ? Is the assignment \(\Omega\mapsto\) N(\(\Omega\)) one to one? 3. Relate \(\Omega\) with N(\(\Omega\)). What can one tell about \(\Omega\) when N(\(\Omega\)) has some special properties? The last problem is connected with the Pompeiu problem. The author treats the case when \(\Omega\) is a strictly convex domain in \({\mathbb{R}}^ n\). The main theorem describes the structure of the null variety at infinity for such \(\Omega\) and the asymptotic behaviour of the function \({\tilde \chi}{}_{\Omega}.\) The generalization to the case of strictly horospherically convex domain \(\Omega\) in the n-dimensional hyperbolic space is also obtained. Reviewer: D.V.Alekseevskij Cited in 1 ReviewCited in 5 Documents MSC: 32B15 Analytic subsets of affine space 32F99 Geometric convexity in several complex variables 42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces) 52A55 Spherical and hyperbolic convexity Keywords:null variety; measurable bounded set; analytic set; Fourier transform; characteristic function; strictly convex domain; strictly horospherically convex domain; hyperbolic space PDFBibTeX XMLCite \textit{T. Kobayashi}, J. Fac. Sci., Univ. Tokyo, Sect. I A 36, No. 3, 389--478 (1989; Zbl 0701.32008)