Aleksander Abanin
+7(863) 2184000
ext.
14021
Head of the department
Institute of Mathematics, Mechanics, and Computer Science named after of I.I. Vorovich
Chairman
Southern Federal University
Research interests:
My research interests are concentrated on representing systems and sufficient sets, generating ideals in weighted spaces of entire functions, the theory of ultradifferentiable functions and ultradistributions, the theory of duality of function spaces, the general theory of weighted spaces of holomorphic functions and operators in them.
Research projects:
1. Representing systems and sufficient sets.
It was undertaken a systematic study of (weakly) sufficient sets in spaces of entire functions of various types and developed some applications to the theory of representing systems and convolution equations (19802016). It was proved that the classes of weakly sufficient and effective sets coincide (1985). It was created a theory of absolutely representing systems of subspaces in spectra of locally convex spaces (19961999; 20052010). There were developed some new methods to study properties of weakly sufficient sets and absolutely representing systems in multidimensional case(198694). A connection between weakly sufficient and sampling sets in the space of all holomorphic functions of polynomial grows in the unit ball and in Hormander algebras of holomorphic functions of a general type was established (20152016). My PhD students in this research field are Yu.S. Nalbandyan (1995), G.A. Semenova (2000), K.A. Mikhailov (2010), and V.A. Varziev (2013).
2. Generating ideals.
There were developed some new methods to study generating ideals in nonradial classes of weighted spaces of entire functions(1995). A geometrical charachterization of zero sets of generating ideals was obtained (1999). My PhD students in this research field are I.S. Shabarshina (2000) and V.V. Shamraeva (2005).
3. Theory of ultradifferentiable functions and ultradistributions.
It was obtained a complete description of spaces of ultradifferentiable functions admitting analogs of Whitney's extension theorem (1999). It was developed a new theory of ultradistributions that contains the all previous classical theories (19972007). My PhD students in this research field are E.S. Tishchenko and E.R. Lyalikova.
4. Convolution equations and division theorems.
There were established some criteria of solvability for a convolution equation in spaces of holomorphic functions of polynomial grows in a convex domain and proved the existance of exponentialpolynomial bases in the kernel of the corresponding operator (20092013; in collaboration with Le Hai Khoi and R. Ishimura). It was established a criterion for the division problem to be valid in a space of entire functions with twoterms growth conditions (2010; in collaboration with D.A. Polyakova).
5. Duality of function spaces.
There were developed new methods to describe conjugate spaces for inductive limits of sequences of Banach spaces of infinitely differentiable functions and projective spectra of such spaces (2006; in collaboration with I.A. Filipiev). A mutual duuality between spaces of holomorphic functions of polynomial growth and Frechet spaces of holomorphic functions of a given boundary smoothness was established (20092013; in collaboration with Le Hai Khoi). My PhD student in this field is S.V. Petrov (2011).
6. General theory of weighted spaces of holomorphic functions.
It was obtained a deep generalization of the wellknown Hormander theorem on continuation of entire functions with growth conditions and developed some of its applications to the description of canonical systems of weights (2010). There were established new criteria for a weighted space of holomorphic functions to belong to the family of compact spectra (20122013). A natural connection between topological and algebraic structures of inductive limits of weighted spaces of holomorphic functions and their projective hulls (2014). There were obtained new criteria for the classical integration and differentiation operators to be bounded in weighted spaces of holomorphic functions (20142016). All these results were obtained in collaboration with my PhD student Pham Trong Tien (2013).
Teaching:

Calculus
This is a standard foursemester course of the Calculus for university students studying at the Departments of Mathematics and Mecanics of the Institute of Mathematics, Mechanics and Computer Science.

Complex Analysis
This is a standard course of the Complex Analysis for university students studying at the Deepartments of Mathematics and Mecanics of the Institute of Mathematics, Mechanics and Computer Science.

Advanced course of Analysis
This twosemester course contains the modern measure and Lebesque integral theories for graduate students.