ON A CARLEMAN PROBLEM IN THE CASE OF A DOUBLY PERIODIC GROUP
Let Г be a doubly periodic group whose fundamental region 𝐷 is a rectangle, in which the ratio of the largest side to the shortest one does not exceed 3. The generating transformations of the group and their inverse transformations induce, on the boundary, an involutive inverse shift, discontinuo...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Petrozavodsk State University
2022-09-01
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| Series: | Проблемы анализа |
| Subjects: | |
| Online Access: | https://issuesofanalysis.petrsu.ru/article/genpdf.php?id=11390&lang=ru |
| Summary: | Let Г be a doubly periodic group whose fundamental
region 𝐷 is a rectangle, in which the ratio of the largest side to
the shortest one does not exceed 3. The generating transformations of the group and their inverse transformations induce, on the
boundary, an involutive inverse shift, discontinuous at the vertices.
We consider a particular case of the Carleman problem for functions that are analytic in 𝐷 (the so-called jump problem). We
show that the regularization of the unknown function suggested
by Torsten Carleman leads to an equivalent regularization of the
problem. For this, we rely on the contraction mapping principle
for Banach spaces and use the theory of Weierstrass elliptic functions. The integral representation was first introduced by Carleman
during his talk at the International Congress of Mathematicians
in Zürich in 1932. However, he did not investigate the Fredholm
integral equation obtained by regularizing the jump problem. In
particular, the question of equivalence of the jump problem and
the corresponding Fredholm equation obtained through the given
representation remained open.
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| ISSN: | 2306-3424 2306-3432 |