Best approximations for the weighted combination of the Cauchy-Szegö kernel and its derivative in the mean
In this paper, we study an extremal problem involving best approximation in the Hardy space $H^1$ on the unit disk $\mathbb D$. Specifically, we consider weighted combinations of the Cauchy-Szegö kernel and its derivative, parameterized by an inner funtion $\varphi$ and a complex number $\lambda$,...
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Oles Honchar Dnipro National University
2024-12-01
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| Series: | Researches in Mathematics |
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| Online Access: | https://vestnmath.dnu.dp.ua/index.php/rim/article/view/439/439 |
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| author | V.V. Savchuk M.V. Savchuk |
| author_facet | V.V. Savchuk M.V. Savchuk |
| author_sort | V.V. Savchuk |
| collection | DOAJ |
| description | In this paper, we study an extremal problem involving best approximation in the Hardy space $H^1$ on the unit disk $\mathbb D$. Specifically, we consider weighted combinations of the Cauchy-Szegö kernel and its derivative, parameterized by an inner funtion $\varphi$ and a complex number $\lambda$, and provide explicit formulas for the best approximation $e_{\varphi,z}(\lambda)$ by the subspace $H^1_0$. We also describe the extremal functions associated with this approximation. Our main result gives the form of $e_{\varphi,z}(\lambda)$ as a function of $\lambda$ and shows that, for a sufficiently large module of $\lambda$, the extremal function is linear in $\lambda$ and unique. We apply this result to establish a sharp inequality for holomorphic functions in the unit disk, leading to a new version of the Schwarz-Pick inequality. |
| format | Article |
| id | doaj-art-1b494bc776444238a26bfbcf1b2a630d |
| institution | Kabale University |
| issn | 2664-4991 2664-5009 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | Oles Honchar Dnipro National University |
| record_format | Article |
| series | Researches in Mathematics |
| spelling | doaj-art-1b494bc776444238a26bfbcf1b2a630d2025-01-05T19:42:24ZengOles Honchar Dnipro National UniversityResearches in Mathematics2664-49912664-50092024-12-0132215516110.15421/242426Best approximations for the weighted combination of the Cauchy-Szegö kernel and its derivative in the meanV.V. Savchuk0https://orcid.org/0000-0002-6713-4471M.V. Savchuk1https://orcid.org/0000-0003-0059-2977Institute of Mathematics of NAS of UkraineNational Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"In this paper, we study an extremal problem involving best approximation in the Hardy space $H^1$ on the unit disk $\mathbb D$. Specifically, we consider weighted combinations of the Cauchy-Szegö kernel and its derivative, parameterized by an inner funtion $\varphi$ and a complex number $\lambda$, and provide explicit formulas for the best approximation $e_{\varphi,z}(\lambda)$ by the subspace $H^1_0$. We also describe the extremal functions associated with this approximation. Our main result gives the form of $e_{\varphi,z}(\lambda)$ as a function of $\lambda$ and shows that, for a sufficiently large module of $\lambda$, the extremal function is linear in $\lambda$ and unique. We apply this result to establish a sharp inequality for holomorphic functions in the unit disk, leading to a new version of the Schwarz-Pick inequality.https://vestnmath.dnu.dp.ua/index.php/rim/article/view/439/439hardy spacecauchy-szegö kernelholomorophic functionbest approximationextremal functionschwarz-pick inequality |
| spellingShingle | V.V. Savchuk M.V. Savchuk Best approximations for the weighted combination of the Cauchy-Szegö kernel and its derivative in the mean Researches in Mathematics hardy space cauchy-szegö kernel holomorophic function best approximation extremal function schwarz-pick inequality |
| title | Best approximations for the weighted combination of the Cauchy-Szegö kernel and its derivative in the mean |
| title_full | Best approximations for the weighted combination of the Cauchy-Szegö kernel and its derivative in the mean |
| title_fullStr | Best approximations for the weighted combination of the Cauchy-Szegö kernel and its derivative in the mean |
| title_full_unstemmed | Best approximations for the weighted combination of the Cauchy-Szegö kernel and its derivative in the mean |
| title_short | Best approximations for the weighted combination of the Cauchy-Szegö kernel and its derivative in the mean |
| title_sort | best approximations for the weighted combination of the cauchy szego kernel and its derivative in the mean |
| topic | hardy space cauchy-szegö kernel holomorophic function best approximation extremal function schwarz-pick inequality |
| url | https://vestnmath.dnu.dp.ua/index.php/rim/article/view/439/439 |
| work_keys_str_mv | AT vvsavchuk bestapproximationsfortheweightedcombinationofthecauchyszegokernelanditsderivativeinthemean AT mvsavchuk bestapproximationsfortheweightedcombinationofthecauchyszegokernelanditsderivativeinthemean |