Best approximations for the weighted combination of the Cauchy--Szeg\"o kernel and its derivative in the mean

Viktor V. Savchuk; Maryna V. Savchuk

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arXiv · 2409.10833

Best approximations for the weighted combination of the Cauchy--Szeg\"o kernel and its derivative in the mean

Viktor V. Savchuk·Maryna V. Savchuk

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Abstract

In this paper, we study an extremal problem concerning best approximation in the Hardy space $H^1$ on the unit disk $\mathbb D$. Specifically, we consider weighted combinations of the Cauchy-Szeg\"o kernel and its derivative, parametrized by an inner function $\varphi$ and a complex number $\lambda$, and provide explicit formula of the best approximation $e_{\varphi,z}(\lambda)$ by the subspace $H^1_0$. We also describe the extremal functions associated with this approximation.

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Viktor V. Savchuk, Maryna V. Savchuk. 2024-09-17. Best approximations for the weighted combination of the Cauchy--Szeg\"o kernel and its derivative in the mean. https://arxiv.org/abs/2409.10833

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