# zbMATH — the first resource for mathematics

Asymptotic behaviour of the null variety for a convex domain in a non- positively curved space form. (English) Zbl 0701.32008
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

##### 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