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 $\lambda$,...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Oles Honchar Dnipro National University
2024-12-01
|
| Series: | Researches in Mathematics |
| Subjects: | |
| Online Access: | https://vestnmath.dnu.dp.ua/index.php/rim/article/view/439/439 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | 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. |
|---|---|
| ISSN: | 2664-4991 2664-5009 |