Point interactions on the line and Riesz bases of δ -functions
We present the description of a relationship between the Sobolev spaces $W^1_2 (R),\; W^2_2 (R)$ and the Hilbert space $\ell_2$. Let $Y$ be a finite or countable set of points on $R$ and let $d := \mathrm{inf} \bigl\{ | y\prime y\prime \prime | , y\prime , y\prime \prime \in Y, y\prime \not = y\...
Збережено в:
| Дата: | 2017 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Українська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2017
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/1808 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507670587375616 |
|---|---|
| author | Kovalev, Yu. G. Ковальов, Ю. Г. |
| author_facet | Kovalev, Yu. G. Ковальов, Ю. Г. |
| author_sort | Kovalev, Yu. G. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:28:05Z |
| description | We present the description of a relationship between the Sobolev spaces $W^1_2 (R),\; W^2_2 (R)$ and the Hilbert space $\ell_2$.
Let $Y$ be a finite or countable set of points on $R$ and let $d := \mathrm{inf}
\bigl\{ | y\prime y\prime \prime | , y\prime , y\prime \prime \in Y, y\prime \not = y\prime \prime \bigr\}$.
By using this
relationship, we prove that if d = 0, then the systems of delta-functions $\bigl\{ \delta (x y_j), y_j \in Y \bigr\} $ and their derivatives
$\bigl\{ \delta \prime (x y_j), y_j \in Y \bigr\} $ do not form Riesz bases in the closures of their linear spans in the Sobolev spaces $W^1_2 (R),\; W^2_2 (R)$ but, conversely, form these bases in the case where $d > 0$. We also present the description of the Friedrichs and
Krein extensions and prove their transversality. Moreover, the construction of a basis boundary triple and the description
of all nonnegative selfadjoint extensions of the operator $A\prime$ are proposed. |
| first_indexed | 2026-03-24T02:13:00Z |
| format | Article |
| fulltext |
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| id | umjimathkievua-article-1808 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-03-24T02:13:00Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/45/179cb3417fd99a8cbe226f55b7104845 |
| spelling | umjimathkievua-article-18082019-12-05T09:28:05Z Point interactions on the line and Riesz bases of δ -functions Точкові взаємодії на прямій і базиси Ріса з δ -функцій Kovalev, Yu. G. Ковальов, Ю. Г. We present the description of a relationship between the Sobolev spaces $W^1_2 (R),\; W^2_2 (R)$ and the Hilbert space $\ell_2$. Let $Y$ be a finite or countable set of points on $R$ and let $d := \mathrm{inf} \bigl\{ | y\prime y\prime \prime | , y\prime , y\prime \prime \in Y, y\prime \not = y\prime \prime \bigr\}$. By using this relationship, we prove that if d = 0, then the systems of delta-functions $\bigl\{ \delta (x y_j), y_j \in Y \bigr\} $ and their derivatives $\bigl\{ \delta \prime (x y_j), y_j \in Y \bigr\} $ do not form Riesz bases in the closures of their linear spans in the Sobolev spaces $W^1_2 (R),\; W^2_2 (R)$ but, conversely, form these bases in the case where $d > 0$. We also present the description of the Friedrichs and Krein extensions and prove their transversality. Moreover, the construction of a basis boundary triple and the description of all nonnegative selfadjoint extensions of the operator $A\prime$ are proposed. Приведено описание некоторой связи пространств Соболева $W^1_2 (R),\; W^2_2 (R)$ и гильбертова пространства $\ell_2$. Пусть $Y$ — конечная или исчислимая монотонная последовательность точек на $R$ и $d := \mathrm{inf} \bigl\{ | y\prime y\prime \prime | , y\prime , y\prime \prime \in Y, y\prime \not = y\prime \prime \bigr\}$. С помощью этой связи доказано, что при условии $d = 0$ системы дельта-функций $\bigl\{ \delta (x yj), y_j \in Y \bigr\} $ и их производных $\bigl\{ \delta \prime (x y_j), y_j \in Y \bigr\} $ не образуют базисы Риса в замыкании своих линейных оболочек в гильбертовых пространствах $W^1_2 (R),\; W^2_2 (R)$, а при условии $d > 0$ — образуют. Дано описание расширений Фридрихса и Крейна, продемонстрирована их трансверсальность, приведены конструкция базисной граничной тройки и описание всех неотрицательных самосопряженных расширений оператора $A^{\prime}$ . Institute of Mathematics, NAS of Ukraine 2017-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1808 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 12 (2017); 1615-1624 Український математичний журнал; Том 69 № 12 (2017); 1615-1624 1027-3190 uk https://umj.imath.kiev.ua/index.php/umj/article/view/1808/790 Copyright (c) 2017 Kovalev Yu. G. |
| spellingShingle | Kovalev, Yu. G. Ковальов, Ю. Г. Point interactions on the line and Riesz bases of δ -functions |
| title | Point interactions on the line and Riesz bases of δ -functions |
| title_alt | Точкові взаємодії на прямій і базиси Ріса з δ -функцій |
| title_full | Point interactions on the line and Riesz bases of δ -functions |
| title_fullStr | Point interactions on the line and Riesz bases of δ -functions |
| title_full_unstemmed | Point interactions on the line and Riesz bases of δ -functions |
| title_short | Point interactions on the line and Riesz bases of δ -functions |
| title_sort | point interactions on the line and riesz bases of δ -functions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1808 |
| work_keys_str_mv | AT kovalevyug pointinteractionsonthelineandrieszbasesofdfunctions AT kovalʹovûg pointinteractionsonthelineandrieszbasesofdfunctions AT kovalevyug točkovívzaêmodíínaprâmíjíbazisirísazdfunkcíj AT kovalʹovûg točkovívzaêmodíínaprâmíjíbazisirísazdfunkcíj |