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Optimization problems related to the John uniqueness theorem. (English. Russian original) Zbl 1215.46028
St. Petersbg. Math. J. 21, No. 5, 705-729 (2010); translation from Algebra Anal. 21, No. 5, 37-69 (2009).
The classical F. John uniqueness theorem states that, if a function \(f\in C^\infty(\mathbb{R}^n)\) with zero integrals over all spheres of a fixed radius \(r\) vanishes in some ball of radius \(r\), then \(f= 0\) in \(\mathbb{R}^n\). Several variations of the classical F. John uniqueness theorem have been developed. One of them refers to the convolution equation, \(f* T= 0\), where \(T\) is a given distribution with compact support in \(\mathbb{R}^n\). This research paper addresses the spectrum structure problem whereby a distribution is periodic in the mean and satisfies uniqueness conditions of the F. John type. The solution of this problem is obtained for a wide class of distributions on arbitrary Riemannian two-point-homogeneous spaces. Many details are clearly included in the paper.

MSC:
46F10 Operations with distributions and generalized functions
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