Entire Functions Theory and Its Applications. Applied Mathematics.

Place of birth — Ekaterinburg (Sverdlovsk), Russia. I have finished Department of Mathematics, Faculty Phisics & Mathematics, Ural State University, Ekaterinburg (Sverdlovsk), Russia; in the same place — Postgraduate School.

• 1965 — Ph. D. in Mathematics, Institute of Mathematics, Siberian Branch, AS USSR, Novosibirsk, Russia, Title of doctoral paper "Some Questions of Growth Theory of Entire Functions of Several Complex Variables", supervisor — Corresponding Member of AS USSR (Academy of Sciences of USSR), Prof. Valentin K. Ivanov .
• 1991 — D. Sc. in Mathematics in the same place. Title of doctoral thesis ‘’Asymptotic Characteristics of Entire Functions and Their Applications’’.

Main places of work:

• 2019– at present – Leading Scientist; Krasnoyarsk Scientific Center of the Siberian Branch of the RAS; .
• 2010–2018— Professor of Chair "Business-Informatics", School of Business Management and Economics, SFU (Siberian Federal University), Krasnoyarsk, Russia.
• 1983–2010 — Associate Professor, Professor of Chair of High Mathematics, Krasnoyarsk State Architecture and Civil Engineering Academy (IAC SFU with 2007), Krasnoyarsk, Russia.
• 1978–1983 — Leader of Mathematical Modelling Groupe, Institute of Medical Problems of North, Siberian Branch, Russian Academy of Medical Sciences, Krasnoyarsk, Russia.
• 1965–1976 — Scientist, Senior Scientist of Functions Theory Laboratory, Institute of Physics, Siberian Branch, Russian Academy of Sciences, Krasnoyarsk, Russia. Simultaneously Associate Professor (half of a day), Chair of Functions Theory, Krasnoyarsk State University.

I have supervised 4 Ph.D. in Mathematics students.

Main Results:

Entire Functions of One Variable. Plane indicator diagram of an entire function (e. f.) of an order ρ ≠ 1 with the indicator of general form, its geometric description with the help of (ρ, α)-convex sets. Generalized Laplace–Borel transform, its role in a description of e. f. spaces [ρ, h], [ρ, h), where h is a given 2π-periodic trigonometrically ρ-convex function (an analogue Polya Theorem) and its some applications in complex analysis. Analytic proximate orders and their generators. Relations between indicator and conjugate diagrams for e. f. with a given proximate order and an nonnegative indicator, and for e. f. of order 1 and minimal type – with any indicator (an analogue Polya Theorem). Application: the criterion of representation of meromorphic function in the form of the sum of the series of principal parts of the Laurent expansion of this function near its poles and an e. f. An analog of the Paley-Wiener Theorem for e. f. belonging to Lp, 1 < p < ∞, p ≠ 2 in $\mathbb{R}$. The construction of the Chebyshev monic polynomial of degree\ $n$\ with zeros on the circle and with the smallest deviation from zero on an arc of the circle. he Puiseux series generated by the power function , where , is considered. A version of the Pólya–Bernstein theorem for an entire function of order $\rho \neq 1$ and normal type is proposed. It is applied to describe the domain of analytic continuation of the Puiseux series generated by the power function .$z = w^{1/\rho}$, $\rho > 0,\ \rho \neq 1$. The domain of summability of a “regular” Puiseux series is found (this is a many-sheeted “Borel polygon”); in the case , the “onesheeted” result of Borel is substantially extended. These results make it possible to describe domains of analytic continuation of the Puiseux expansions of popular many-sheeted functions (such as inverses of rational functions). A new approach to the solution of algebraic equations is found.

Entire Functions of Many Variables. Development of new growth characteristics of entire functions of many variables (e. f. m. v.): the function of orders, the system of type-functions of e. f. m. v. in the directions of its nonzero order growth, their properties, structure, the relation with Taylor coefficients of its expansion in the multiple power series. Existence of e. f. m. v. with given mentioned growth characteristics. Сomparative growth of the maximum of the modulus in a polydisk and the maximal term of an e. f. m. v. The system of regularized indicators of e. f. m. v. of a finite nonzero order in the totality of variables. Indicator diagram of an e. f. m. v. with prescribed system of positive conjugate orders, associated with its indicator. Generalized Laplace--Borel transform of this function; its role in a description of e. f. m. v. space [ρ, h], where h is a given plurisubharmonic function with properties of the indicator (an analog of the Polya--Martineau--Ehrenprice Theorem; applications: the system of integral summation methods for multiple power series, existence of e. f. m. v. with a given positive indicator). Extensions of the class of entire functions of e. f. m. v.: the functions equivalent to e. f. several variables, holomorphic functions in a~multidimensional torus; the description of these functions in terms of geometric properties of their Laurent series supports; growth characteristics and their properties. Preliminary results: the modification of an analog of Abels lemma and an~analog of the Cauchy--Hadamard formula for multiple Laurent series, a~multidimensional analog of the expansion of a~holomorphic function in a~Laurent series.

Extrapolation Entire Functions from a Finite Set. The recovery operator of solutions of homogeneous OLDE (ordinary linear differential equations) of fixed order with unknown constant coefficients their values at a finite number of nodes of a uniform net (the Prony algorithm), conditions for their existence, uniquess, stability. Its specific character in the case of wave-types solutions of these equations. Applications in biophysics: mathematical prediction of behavior dynamics of the sugar curves. Modified Prony Algorithm for an Inhomogeneous OLDE with сonstant сoefficients. The variant of Carlson Theorem for quasipolynomials. An inverse problem for an homogeneous evolutionary differential equation of finite order with e. f. m. v. coefficients (the multidimensional version of the Prony Algorithm). An optimal error for extrapolation of e. f. m. v. in the given point from a finite set in the Wiener class and its estimate. Optimal error of analytic continuation from a finite set with inaccurate data in Hilbert spaces of holomorphic functions with reproducing kernel.

Mathematical Methods in Economics. The mathematical model of optimal distribution of limited resource of social-economic contents between people groups under different conditions. It demonstrates how the model can be used to possibly solve the problem of objectification of greenhouse gas emission quotas. A mathematical model of the just distribution of bonus funds.

1. L. S. Maergoiz, Asymptotic Characteristics of Entire Functions and Their Applications in Mathematics and Biophysics, Second edition (revised and enlarged), Kluwer Academic Publishers, Dordrecht/Boston/London, 2003, 362 pp.    
2. L. S. Maergoiz, “Indicator Diagram and Generalized Borel-Laplace Transforms for Entire Functions of a Given Proximate Order”, Algebra i Analiz, 12:2 (2000), 1–63  ; St. Peterburg Math. J., 12:2 (2001), 191–232  
3. L. S. Maergoiz, Doklady Mathematics, 78:3 (2008), 822–824  
4. L. S. Maergoiz, N. N. Rybakova, “Chebyshev polynomials with zeros on a circle and adjacent problems”, Algebra i Analiz, 25:6 (2014), 965–979
5. L. S. Maergoiz, “Extensions of the class of entire functions of several variables and related topics”, Sibirsk. Mat. Zh., 55:5 (2014), 1137-1158  ; Siberian Math. J., 41:6 (2000), 1126–1136      

• Krasnoyarsk Scientific Center of SB RAS
• Institute of Economics, Management and Environmental Studies, Siberian Federal University, Krasnoyarsk
• Institute of Urban Construction, Management and Regional Economics, Siberian Federal University, Krasnoyarsk
• Krasnoyarsk State Academy of Architecture and Construction