Inverse problems, Sobolev–Chebyshev polynomials and asymptotics
UDC 517.9 Let $(u,v)$ be a pair of quasidefinite and symmetric linear functionals with $\{P_{n}\}_{n\geq0}$ and $\{Q_{n}\}_{n\geq0}$ as respective sequences of monic orthogonal polynomial (SMOP). We define a sequence of monic polynomials $\{R_{n}\}_{n\geq0}$ as follows: $$\...
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| Datum: | 2023 |
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Institute of Mathematics, NAS of Ukraine
2023
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/7293 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512646938230784 |
|---|---|
| author | Molano, Luis Alejandro Molano Molano, Luis Alejandro Molano |
| author_facet | Molano, Luis Alejandro Molano Molano, Luis Alejandro Molano |
| author_sort | Molano, Luis Alejandro Molano |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2024-06-19T00:34:48Z |
| description | UDC 517.9
Let $(u,v)$ be a pair of quasidefinite and symmetric linear functionals with $\{P_{n}\}_{n\geq0}$ and $\{Q_{n}\}_{n\geq0}$ as respective sequences of monic orthogonal polynomial (SMOP). We define a sequence of monic polynomials $\{R_{n}\}_{n\geq0}$ as follows: $$\frac{P_{n+2}'(x)}{n+2}+b_{n}\frac{P_{n}'(x)}{n}-Q_{n+1}(x)=d_{n}R_{n-1}(x),\quad n\geq1.$$
We give necessary and sufficient conditions for $\{R_{n}\}_{n\geq0}$ to be orthogonal with respect to a quasidefinite linear functional $w.$   In addition, we consider the case where $\{P_{n}\}_{n\geq0}$ and $\{Q_{n}\}_{n\geq0}$ are  monic Chebyshev polynomials of the first and second kinds, respectively, and study the relative outer asymptotics of Sobolev polynomials orthogonal with respect to the Sobolev inner product\begin{equation*}\langle p,q\rangle _{S}=\int\limits _{-1}^{1}pq(1-x^{2})^{-1/2}dx+\lambda_{1}\int\limits _{-1}^{1}p'q'(1-x^{2})^{1/2}dx+\lambda_{2}\int\limits _{-1}^{1}p''q''d\mu(x),\end{equation*} where $\mu$ is a positive Borel measure associated with $w$ and $\lambda_{1},\lambda_{2}>0,$ $\lambda_{2}$ is a linear polynomial of $\lambda_{1}.$
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| doi_str_mv | 10.3842/umzh.v75i10.7293 |
| first_indexed | 2026-03-24T03:32:06Z |
| format | Article |
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| id | umjimathkievua-article-7293 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:32:06Z |
| publishDate | 2023 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
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| spelling | umjimathkievua-article-72932024-06-19T00:34:48Z Inverse problems, Sobolev–Chebyshev polynomials and asymptotics Inverse problems, Sobolev–Chebyshev polynomials and asymptotics Molano, Luis Alejandro Molano Molano, Luis Alejandro Molano Orthogonal polynomials, Inverse problems, Sobolev-Tchebichef polynomials, Asymptotics UDC 517.9 Let $(u,v)$ be a pair of quasidefinite and symmetric linear functionals with $\{P_{n}\}_{n\geq0}$ and $\{Q_{n}\}_{n\geq0}$ as respective sequences of monic orthogonal polynomial (SMOP). We define a sequence of monic polynomials $\{R_{n}\}_{n\geq0}$ as follows: $$\frac{P_{n+2}'(x)}{n+2}+b_{n}\frac{P_{n}'(x)}{n}-Q_{n+1}(x)=d_{n}R_{n-1}(x),\quad n\geq1.$$ We give necessary and sufficient conditions for $\{R_{n}\}_{n\geq0}$ to be orthogonal with respect to a quasidefinite linear functional $w.$   In addition, we consider the case where $\{P_{n}\}_{n\geq0}$ and $\{Q_{n}\}_{n\geq0}$ are  monic Chebyshev polynomials of the first and second kinds, respectively, and study the relative outer asymptotics of Sobolev polynomials orthogonal with respect to the Sobolev inner product\begin{equation*}\langle p,q\rangle _{S}=\int\limits _{-1}^{1}pq(1-x^{2})^{-1/2}dx+\lambda_{1}\int\limits _{-1}^{1}p'q'(1-x^{2})^{1/2}dx+\lambda_{2}\int\limits _{-1}^{1}p''q''d\mu(x),\end{equation*} where $\mu$ is a positive Borel measure associated with $w$ and $\lambda_{1},\lambda_{2}>0,$ $\lambda_{2}$ is a linear polynomial of $\lambda_{1}.$   УДК 517.9 Обернені задачі, поліноми Соболєва–Чебишова та асимптотика Нехай $(u,v)$ – пара квазівизначених симетричних лінійних функціоналів, в яких $\{P_{n}\}_{n\geq0}$ і $\{Q_{n}\}_{n\geq0}$ є відповідними послідовностями монічних ортогональних поліномів (ПМОП). Послідовність монічних поліномів $\{R_{n}\}_{n\geq0}$ визначено таким чином: $$\frac{P_{n+2}'(x)}{n+2}+b_{n}\frac{P_{n}'(x)}{n}-Q_{n+1}(x)=d_{n}R_{n-1}(x),\quad n\geq1.$$ Наведено необхідні та достатні умови для того, щоб послідовність  $\{R_{n}\}_{n\geq0}$ була ортогональною до квазівизначеного лінійного функціонала $w.$  Крім того, розглянуто випадок, коли $\{P_{n}\}_{n\geq0}$ і $\{Q_{n}\}_{n\geq0}$ – монічні поліноми Чебишова першого і другого роду відповідно, та вивчено відносну зовнішню асимптотику поліномів Соболєва, ортогональних щодо соболєвського скалярного добутку  \begin{equation*}\langle p,q\rangle _{S}=\int\limits _{-1}^{1}pq(1-x^{2})^{-1/2}dx+\lambda_{1}\int\limits _{-1}^{1}p'q'(1-x^{2})^{1/2}dx+\lambda_{2}\int\limits _{-1}^{1}p''q''d\mu(x),\end{equation*} де $\mu$ – додатна борелівська міра, пов’язана з $w$ і $\lambda_{1},\lambda_{2}>0,$ $\lambda_{2}$ – лінійний поліном від $\lambda_{1 }.$ Institute of Mathematics, NAS of Ukraine 2023-10-24 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/7293 10.3842/umzh.v75i10.7293 Ukrains’kyi Matematychnyi Zhurnal; Vol. 75 No. 10 (2023); 1411 - 1428 Український математичний журнал; Том 75 № 10 (2023); 1411 - 1428 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7293/9908 Copyright (c) 2023 Alejandro Molano |
| spellingShingle | Molano, Luis Alejandro Molano Molano, Luis Alejandro Molano Inverse problems, Sobolev–Chebyshev polynomials and asymptotics |
| title | Inverse problems, Sobolev–Chebyshev polynomials and asymptotics |
| title_alt | Inverse problems, Sobolev–Chebyshev polynomials and asymptotics |
| title_full | Inverse problems, Sobolev–Chebyshev polynomials and asymptotics |
| title_fullStr | Inverse problems, Sobolev–Chebyshev polynomials and asymptotics |
| title_full_unstemmed | Inverse problems, Sobolev–Chebyshev polynomials and asymptotics |
| title_short | Inverse problems, Sobolev–Chebyshev polynomials and asymptotics |
| title_sort | inverse problems, sobolev–chebyshev polynomials and asymptotics |
| topic_facet | Orthogonal polynomials Inverse problems Sobolev-Tchebichef polynomials Asymptotics |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7293 |
| work_keys_str_mv | AT molanoluisalejandromolano inverseproblemssobolevchebyshevpolynomialsandasymptotics AT molanoluisalejandromolano inverseproblemssobolevchebyshevpolynomialsandasymptotics |