Beijing-Saint Petersburg Mathematics Colloquium (online)
Abstract
The definition of all objects used in this announcement will be given in the talk. Let an analytical function f belong to the Hardy class Hp in the unit disc D. Then f may be represented as the product, f=IOf, where I is the so-called inner function, it means that |I(z)|<1 for z belonging D, and |I(z)|=1 for almost every z on the unit circle, and the so-called outer function Of is defined by values of |f(z)| on the unit circle. One of those two factors may be absent. The cited statement is the classical result, the factors I and Of are in a sense independent for the functions f from Hp. Let us consider a class X which is contained in H1 and consists of functions f continuous in the closed disc D--. Then f=IOf ,the inner function I is in general discontinuous in D-- what implies that the outer function Of is to compensate the points of discontinuity of the inner function I. The talk is devoted to the concrete way of this compensation and to the specific properties of outer functions belonging to the analytical Holder classes and to the classes of functions of variable smoothness. The consequences about the half-smoothness of an analytical function in comparison with the smoothness of its modulus on the boundary will be given too.
Bio
Professor Nikolai A. Shirokov is a worldwide renowned expert in harmonic and complex analysis which includes the theory of approximations, boundary properties of holomorphic functions and the theory of singular integrals. Having graduated from Saint Petersburg State University in 1971, he has got his Ph.D.in 1973, became Doctor of Science (Saint Petersburg department of Steklov Institute) in 1985 and got his Professor degree in 1988. Professor Shirokov is the author of 82 research publications (Scopus) and a supervisor of a big number of students (both graduate and undergraduate). In 2014 he was officially recognized as ‘the best teacher’ of Higher School of Economics which is one of the most prestigious Russian Universities. N.A. Shirokov heads the department of Math.Analysis of Saint Petersburg State University since 2005 and the Department of Applied Mathematics at Higher School of Economics (Saint Petersburg branch) since 2013.
