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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
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