01.01.01 (Real analysis, complex analysis, and functional analysis)
theory of duality; ultradifferentiable functions and ultradistributions; Fourier transformation; representing systems; entire functions; extension of infinitely differentiable functions by Whitney.
Some new criteria for weakly sufficient sets in weighted spaces of holomorphic functions were obtained. A connection between weakly sufficient sets and effective sets in the sense of G. Iyer was completely investigated. Geometric type criteria for representation of holomorphic functions by series of generalized exponentials were obtained. A complete description of generating ideals in rings of entire functions was found in terms of their generators (with I. S. Shabarshina). A criterion that spaces of ultradifferentiable functions of Beurling or of Roumier type admit an analog of Whitney's extension theorem was obtained.
Graduated from Faculty of Mechanics and Mathematics of Rostov State University (RSU) in 1977 (department of the theory of functions and functional analysis). Ph.D. thesis was defended in 1981. D.Sci. thesis was defended in 1995. A list of my works contains more than 100 titles. Since 1995 I and Yu. F. Korobeinik have led the research seminar at RSU on linear problems in analysis.
Abanin A. V. On Whitney's extension theorem for spaces of ultradifferentiable functions // Math. Ann., 2001, 320, 115–126.
Abanin A. V., Shabarshina I. S. Zero sets of generators and differential ideals // Math. Nachr., 2002, 238, 5–15.
Alexey Borisovich Shabat (on his 80's anniversary) A. V. Abanin, S. N. Askhabov, A. B. Borisov, R. A. Bostanov, A. V. Zhiber, V. E. Zakharov, S. B. Klimentov, Yu. F. Korobeinik, A. G. Kusraev, S. S. Kutateladze, A. V. Mikhailov, A. P. Soldatov, Kh. G. Umarov, S. M. Umarkhadzhiev Vladikavkaz. Mat. Zh., 19:3 (2017), 83–85
Shabat Aleksei Borisovich (on his 75th birthday) A. V. Abanin, S. N. Askhabov, A. B. Borisov, R. A. Bostanov, A. V. Zhiber, V. E. Zakharov, S. B. Klimentov, Yu. F. Korobeinik, A. G. Kusraev, S. S. Kutateladze, A. V. Mikhailov, A. P. Soldatov, Kh. G. Umarov, S. M. Umarkhadzhiev Vladikavkaz. Mat. Zh., 14:2 (2012), 71–73