The gleason distance РАССТОЯНИЕ ГЛИСОНА

• Main Author: Ovchintsev Mikhail Petrovich
• Format: Article
• Language: English
• Published: Moscow State University of Civil Engineering (MGSU) 2013-08-01
• Series: Vestnik MGSU
• Subjects: Gleason distance; Mobius transformation; conformal mapping; simply connected domain; extremum function; расстояние Глисона; преобразование Мебиуса; конформное отображение; экстремальная функция; односвязная область
• Online Access: http://vestnikmgsu.ru/files/archive/RUS/issuepage/2013/8/4.pdf

First, some basic concepts are considered in the paper, including the Mobius transformation, the unit ball in the space of related analytical functions in the unit circle, and the Gleason distance. The author proves a theorem (demonstrated without any proof) that makes it possible to calculate the Gleason distance between the two opposite points in the pre-set unit circle. The extremum feature appears in the calculation of the Gleason distance, which coincides with the identity map of the unit circle. The Gleason distance between the two points coincides with the regular Euclidean distance between these points. Further, the author considers the Gleason distance in the simply connected domain. The simply connected domain is conformally represented in the unit circle. The two points in the simply connected domain are represented as the corresponding points in the unit circle. The author has proven that the Gleason distance between the two points in the simply connected domain coincide with the Gleason distance between two corresponding points in the unit circle. Then, the author presents a lemma (a statement without proof). It is applied to the problem of the Gleason distance between the two points in the simply connected domain. Next, the author presents several special cases: the Gleason distance as calculated between the two points in the unit circle and between the two points in the upper half-space. The two points are located (with both points being positive numbers) in the unit circle.<br>Приведена теорема для вычисления расстояния Глисона между двумя противоположными точками, лежащими в единичном круге, а также лемма о получении экстремальной функции в этой задаче. Разобраны частные случаи вычисления расстояния Глисона в единичном круге и в верхней полуплоскости.