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...
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| Format: | Article |
| Language: | English |
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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 |
| _version_ | 1818564341123776512 |
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| author | F. N. Garif’yanov E. V. Strezhneva |
| author_facet | F. N. Garif’yanov E. V. Strezhneva |
| author_sort | F. N. Garif’yanov |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-14T01:27:45Z |
| format | Article |
| id | doaj.art-5ae2bbba2fa240068b67bea72ef4005e |
| institution | Directory Open Access Journal |
| issn | 2306-3424 2306-3432 |
| language | English |
| last_indexed | 2024-12-14T01:27:45Z |
| publishDate | 2020-06-01 |
| publisher | Petrozavodsk State University |
| record_format | Article |
| series | Проблемы анализа |
| spelling | doaj.art-5ae2bbba2fa240068b67bea72ef4005e2022-12-21T23:22:08ZengPetrozavodsk State UniversityПроблемы анализа2306-34242306-34322020-06-019 (27)2586710.15393/j3.art.2020.7930THE SUMMARY EQUATION FOR FUNCTIONS ANALYTICAL OUTSIDE FOUR SQUARES. APPLICATIONSF. N. Garif’yanov0E. V. Strezhneva1Kazan State Power Engineering UniversityKazan National Research Technical University named after A. N. TupolevWe 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.http://issuesofanalysis.petrsu.ru/article/genpdf.php?id=7930&lang=ruequivalent regularizationcarleman problemmoments of entire functions |
| spellingShingle | F. N. Garif’yanov E. V. Strezhneva THE SUMMARY EQUATION FOR FUNCTIONS ANALYTICAL OUTSIDE FOUR SQUARES. APPLICATIONS Проблемы анализа equivalent regularization carleman problem moments of entire functions |
| title | THE SUMMARY EQUATION FOR FUNCTIONS ANALYTICAL OUTSIDE FOUR SQUARES. APPLICATIONS |
| title_full | THE SUMMARY EQUATION FOR FUNCTIONS ANALYTICAL OUTSIDE FOUR SQUARES. APPLICATIONS |
| title_fullStr | THE SUMMARY EQUATION FOR FUNCTIONS ANALYTICAL OUTSIDE FOUR SQUARES. APPLICATIONS |
| title_full_unstemmed | THE SUMMARY EQUATION FOR FUNCTIONS ANALYTICAL OUTSIDE FOUR SQUARES. APPLICATIONS |
| title_short | THE SUMMARY EQUATION FOR FUNCTIONS ANALYTICAL OUTSIDE FOUR SQUARES. APPLICATIONS |
| title_sort | summary equation for functions analytical outside four squares applications |
| topic | equivalent regularization carleman problem moments of entire functions |
| url | http://issuesofanalysis.petrsu.ru/article/genpdf.php?id=7930&lang=ru |
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