L.I. Ronkin (1931- 1998)

L.I. Ronkin was born on January 2, 1931 in Kharkiv. Being a student of senior classes of high school he was fond of mathematics. With some fellow students, among whom there were Yu. Lyubich, M. Azbel, V. Maleev, who later became famous mathematicians and physicists, he organized a mathematical circle, where they independently studied topics that went beyond the scope of school curriculum.

In 1948, Lev Isaakovich entered the Faculty of Physics and Mathematics of Kharkiv University (V.N. Karazin Kharkiv National University now). Those years, the range of his hobbies was very wide: mathematics, acrobatics, rock climbing. In his third year of university, he took a special course "Almost Periodic Functions" taught by B.Ya. Levin, and Boris Yakovlevich proposed to him to solve a problem related to the Levitan almost periodic functions. Having done a research, not lately than in 1953, L. I. Ronkin published the paper “On the Approximation of Entire Functions by Trigonometric Polynomials” (DAN SSSR, 1953). He obtained the results on entire functions of one and several variables that have not been surpassed yet and are still used by specialists all over the world.

After graduating from university, Lev Isaakovich was sent to Minsk to work as a teacher at a night school. Working at school, he went on actively doing research in mathematics and permanently stayed in touch with B.Ya. Levin, who got him interested in a number of questions related to entire functions of several complex variables.

In 1958, returning to Kharkiv, L. I. Ronkin began teaching mathematics at Kharkiv Aviation Institute, and the same year he defended his Ph.D. thesis "Entire Functions of Finite Degree of Several Variables." Continuing work in this direction, Lev Isaakovich obtained a number of fundamental results on growth and zero sets of entire functions of several variables and, in fact, he became one of the founders of the multidimensional theory of entire functions. He introduced such characteristics of growth as hypersurfaces of conjugate orders and types, and he showed that they can be applied to functions from a wider class than entire ones. For studying growth and distribution of zeros in each of the variable, the others being fixed, L. I. Ronkin developed a general approach by means of theory of plurisubharmonic functions and potential theory. This allowed him to obtain sharp and complete results on the exceptional sets of drop of the order and of the exponent of convergence of the zeros for the case n = 2 (n being the dimension of the space). For n > 2, the lack of suitable "thinness" characteristics of multidimensional sets led him to the construction of the Γ-capacity, a new capacitary characteristic, which for the first time made it possible to obtain results on the drop sets also for n > 2.

In 1967, Lev Isaakovich defended his doctoral dissertation, the results of which later formed the basis of his monograph "Introduction to the Theory of Entire Functions of Several Variables" (Nauka, 1971). It was translated in 1974 in the United States and since then has been widely used by mathematicians working in the field of multidimensional function theory.

For almost thirty years, since 1969, L. I. Ronkin worked at the Department of Function Theory of B. Verkin Institute for Low Temperatures Physics and Engineering of the National Academy of Sciences of Ukraine. During those years, Lev Isaakovich obtained fundamental results in a number of topics of multidimensional complex analysis. He studied discrete sets of uniqueness for entire functions of several variables and completeness of exponential systems, established important facts on separately analytic functions, proved deep theorems on interpolation of entire functions from algebraic and pseudo-algebraic sets, on convergence of Monge-Ampère currents, and on the multidimensional Nevanlinna theory.

In the mid-1970s, L. I. Ronkin began studying functions of completely regular growth of several variables. He was the first to understand importance of the weak convergence in spaces of generalized functions for the study of the asymptotic behavior and distribution of zeros of analytic functions. The method of the weak convergence has turned out to be fruitful for various classes of functions: entire and holomorphic in one and several complex variables, plurisubharmonic and subharmonic in the whole space and in a cone. The results obtained by L.I. Ronkin, as well as his students and colleagues from different countries, are contained in his book "Functions of Completely Regular Growth" published in 1992 in the Netherlands.

The last ten years of his life Lev Isaakovich devoted to the study of almost periodicity. As a result of this activity, he conceived and started a book under the preliminary title "Almost Periodic Objects of Complex Analysis", which he continued writing the last day of his life. The weak convergence method has proven to be extremely effective in investigating these questions. It allowed L. I. Ronkin to introduce and study such general concepts as almost periodic currents, divisors and holomorphic chains, as well as to study distribution of zero sets of holomorphic almost periodic mappings and distinguish classes of almost periodic divisors that are realized as divisors of almost periodic holomorphic functions. In the paper published in 2001, he introduced an object that later became, under the name ‘the Ronkin function’, a basic tool for idempotent (tropical) analysis.

Lev Isaakovich combined an active scientific work with teaching at the Department of Theory of Functions of Kharkiv University. His remarkable lectures were published in as a textbook "Elements of the Theory of Analytic Functions of Several Variables", Kyiv, 1977. The talent of a teacher allowed him to make - for the first time in world literature - such material interesting for both mathematicians and engineers. For many years he led seminars on multivariate complex analysis and educated many actively working mathematicians.

L. I. Ronkin is the author of more than 80 scientific papers. The original ideas and methods of these works are widely used by mathematicians from different countries.

Eight people wrote and successfully defended candidate and doctoral dissertations under his supervision.

Being a great connoisseur and lover of history and literature, an interesting interlocutor, Lev Isaakovich always attracted people, charging them with his optimism and energy. Among his numerous hobbies, photography took a special place; he mastered this art almost to a professional level. He made a unique collection of photographs of mathematicians of the second half of the twentieth century.