ON THE DIFFERENCE EQUATION ASSOCIATED WITH THE DOUBLY PERIODIC GROUP AND ITS APPLICATIONS
Let D be a rectangle. We consider a four-element linear difference equation defined on D. The shifts of this equation are the generating transformations of the corresponding doubly periodic group and their inverse transformations. We search for a solution in the class of functions that are holomor...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Petrozavodsk State University
2020-12-01
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| Series: | Проблемы анализа |
| Subjects: | |
| Online Access: | https://issuesofanalysis.petrsu.ru/article/genpdf.php?id=9090&lang=ru |
| Summary: | Let D be a rectangle. We consider a four-element
linear difference equation defined on D. The shifts of this equation
are the generating transformations of the corresponding doubly periodic group and their inverse transformations. We search for a solution in the class of functions that are holomorphic outside D and
vanish at infinity. Their boundary values satisfy a H¨older condition
on any compact that does not contain the vertices. At the vertices,
we allow, at most, logarithmic singularities. The independent term
is holomorphic on D, and its boundary value satisfies a H¨older condition. The independent term may not be analytically continuable
across an interval of the boundary, since the solution and the independent term belong to different classes of analytical functions. We
regularize the difference equation and determine the conditions for
the regularization to be equivalent. If the independent term is an
odd function, then the problem is solvable. Additionally, we give
some applications of the difference operator to interpolation problems for integer functions of exponential type and the construction
of biorthogonally conjugated systems of analytical functions |
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| ISSN: | 2306-3424 2306-3432 |