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

TL;DR

This paper investigates the optimal approximation of weighted combinations of the Cauchy-Szeg"o kernel and its derivative in the Hardy space, providing explicit formulas and extremal functions.

Contribution

It introduces explicit formulas for best approximations of weighted Cauchy-Szeg"o kernels and their derivatives in $H^1$, along with associated extremal functions.

Findings

Explicit formula for best approximation $e_{,z}()$ provided.

Characterization of extremal functions in the approximation problem.

Analysis of approximation behavior depending on parameters.

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.