THE SUMMARY EQUATION FOR FUNCTIONS ANALYTICAL OUTSIDE FOUR SQUARES. APPLICATIONS
We consider the lacunary Stieltjes moment problem ∞ ∫𝐹(𝑥)𝑥^(4𝑛+1) exp(−𝑥) 𝑑𝑥 = 𝛽_𝑛, 𝑛 = 0, 1, 2. 0 We search for a solution in the class of entire functions of the exponential type that satisfy the condition 𝐹(𝑖𝑧) = 𝐹(𝑧). Their indicator diagram is a certain octagon. We construct nontrivial sol...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Petrozavodsk State University
2020-06-01
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| Series: | Проблемы анализа |
| Subjects: | |
| Online Access: | http://issuesofanalysis.petrsu.ru/article/genpdf.php?id=7930&lang=ru |
| Summary: | We consider the lacunary Stieltjes moment problem
∞
∫𝐹(𝑥)𝑥^(4𝑛+1) exp(−𝑥) 𝑑𝑥 = 𝛽_𝑛, 𝑛 = 0, 1, 2.
0
We search for a solution in the class of entire functions of the exponential type that satisfy the condition 𝐹(𝑖𝑧) = 𝐹(𝑧). Their
indicator diagram is a certain octagon. We construct nontrivial solutions to the corresponding homogeneous problem. The problem
reduces studying a linear summary equation in the class of functions holomorphic outside four squares. At infinity, they have a
zero of multiplicity at least three. The 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.
We search for a solution in the form of a Cauchy-type integral
with an unknown density over the boundary of those squares. We
suggest a method for the regularization of the summary equation.
An equivalence condition for this regularization is established. Additionally, we identify some special cases, in which the obtained
Fredholm equation of the second kind is solvable. For this, we use
the contraction mapping theorem in a Banach space. |
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| ISSN: | 2306-3424 2306-3432 |