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## Archive of Issues

Uzbekistan Tashkent
Year
2021
Volume
31
Issue
2
Pages
296-310
 Section Mathematics Title Holomorphic continuation into a matrix ball of functions defined on a piece of its skeleton Author(-s) Khudayberganov G.a, Abdullayev J.Sh.a Affiliations National University of Uzbekistana Abstract The question of the possibility of holomorphic continuation into some domain of functions defined on the entire boundary of this domain has been well studied. The problem of describing functions defined on a part of the boundary that can be extended holomorphically into a fixed domain is attracting more interest. In this article, we reformulate the problem under consideration: Under what conditions can we extend holomorphically to a matrix ball the functions given on a part of its skeleton? We describe the domains into which the integral of the Bochner—Hua Luogeng type for a matrix ball can be extended holomorphically. As the main result, we present the criterion of holomorphic continuation into a matrix ball of functions defined on a part of the skeleton of this matrix ball. The proofs of several results are briefly presented. Some recent advances are highlighted. The results obtained in this article generalize the results of L.A. Aizenberg, A.M. Kytmanov and G. Khudayberganov. Keywords matrix ball, Shilov's boundary, Bochner–Hua Luogeng integral, Hardy space, holomorphic continuation, orthonormal system UDC 517.55 MSC 32A10, 32A26, 32A40, 32M15, 46E20 DOI 10.35634/vm210210 Received 27 September 2020 Language English Citation Khudayberganov G., Abdullayev J.Sh. Holomorphic continuation into a matrix ball of functions defined on a piece of its skeleton, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2021, vol. 31, issue 2, pp. 296-310. References Aizenberg L.A. Formuly Karlemana v kompleksnom analize (Carleman formulas in complex analysis), Novosibirsk: Science, 1990. 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