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## Archive of Issues

Russia Izhevsk
Year
2013
Issue
4
Pages
36-54
 Section Mathematics Title The uniform approximation of recurrent functions and almost recurrent functions Author(-s) Danilov L.I.a Affiliations Physical Technical Institute, Ural Branch of the Russian Academy of Sciencesa Abstract We consider the classes of functions $f:{\mathbb R}\to U,$ taking values in a metric space $(U,\rho ),$ which have Bochner transforms from the classes of recurrent functions and almost recurrent functions. We improve the preceding results on the uniform approximation of functions from classes under consideration by elementary functions from the same classes. These results can be applied to the investigation of the problem of the existence of almost recurrent selections for multivalued maps. The selections are supposed to satisfy a number of additional conditions. In the last section of the paper the variant of Lusin's theorem for recurrent functions is proved. Keywords recurrent function, selection, multivalued mapping, Lusin's theorem UDC 517.518.6 MSC 42A75, 54С65 DOI 10.20537/vm130405 Received 30 October 2013 Language Russian Citation Danilov L.I. The uniform approximation of recurrent functions and almost recurrent functions, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2013, issue 4, pp. 36-54. References Danilov L.I. Recurrent and almost recurrent multivalued maps and their selections, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2011, no. 2, pp. 19-51. Danilov L.I. Recurrent and almost recurrent multivalued maps and their selections. II, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2012, no. 4, pp. 3-21. Rokhlin V.A. On the decomposition of a dynamical system into transitive components, Matematicheskii Sbornik, 1949, vol. 25 (67), no. 2, pp. 235-249. Danilov L.I. Almost periodic selections of multivalued mappings, Izv. Otd. Mat. Inform. Udmurt. Gos. Univ., 1993, no. 1, pp. 16-78. Danilov L.I. Measure-valued almost periodic functions and almost periodic selections of multivalued maps, Sbornik: Mathematics, 1997, vol. 188, no. 10, pp. 1417-1438. Danilov L.I. On almost periodic multivalued maps, Mathematical Notes, 2000, vol. 68, no. 1, pp. 71-77. Danilov L.I. Uniform approximation of Stepanov almost periodic functions, Izv. Inst. Mat. Inform. Udmurt. Gos. Univ., 2004, no. 1 (29), pp. 33-48. Danilov L.I. On almost periodic selections of multivalued maps, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2008, no. 2, pp. 34-41. Dolbilov A.M., Shneiberg I.Ya. Multivalued almost periodic mappings and selections of them, Siberian Math. J., 1991, vol. 32, no. 2, pp. 326-328. Fryszkowski A. Continuous selections for a class of non-convex multivalued maps, Studia Math., 1983, vol. 76, no. 2, pp. 163-174. Danilov L.I. On Weyl almost periodic selections of multivalued maps, J. Math. Anal. Appl., 2006, vol. 316, no. 1, pp. 110-127. Danilov L.I. Weyl almost periodic selections of multivalued maps, Izv. Inst. Mat. Inform. Udmurt. Gos. Univ., 2006, no. 3 (37), pp. 27-28. Danilov L.I. On a class of Weyl almost periodic selections of multivalued maps, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2009, no. 1, pp. 24-45. Danilov L.I. On Besicovich almost periodic selections of multivalued maps, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2008, no. 1, pp. 97-120. Nemytskii V.V., Stepanov V.V. Kachestvennaya teoriya differentsial'nykh uravnenii (Qualitative theory of differential equations), Moscow-Izhevsk: Regular and Chaotic Dynamics, 2004, 456 p. Anosov D.V., Aranson S.Kh., Arnold V.I., Bronshtein I.U., Grines V.Z., Ilyashenko Yu.S. Ordinary differential equations and smooth dynamical systems, Berlin-Heidelberg-New York: Springer-Verlag, 1997. Danilov L.I. On uniform approximation of Weyl and Besicovich almost periodic functions, Izv. Inst. Mat. Inform. Udmurt. Gos. Univ., 2006, no. 1 (35), pp. 33-48. Lyusternik L.A., Sobolev V.I. Kratkii kurs funktsional'nogo analiza (A Short course of functional analysis), Moscow: Vysshaya shkola, 1982, 271 p. Danilov L.I. On the uniform approximation of a function that is almost periodic in the sense of Stepanov, Izvestiya Vysshikh Uchebnykh Zavedenii, Matematika, 1998, no. 5, pp. 10-18. Iosida K. Functional analysis, New York: Springer, 1965. Translated under the title Funktsional'nyi analiz, Moscow: Mir, 1967, 624 p. Full text