An extremal problem for characteristic functions

TL;DR

This paper investigates the extremal problem of approximating characteristic functions of subsets of the unit circle by functions in a shifted Hardy algebra, using operator theory and conformal mappings.

Contribution

It provides explicit solutions to the extremal problem in key cases, employing Toeplitz, Hankel operators, and conformal mapping techniques.

Findings

Explicit solutions for specific cases of the extremal problem

Application of Toeplitz and Hankel operator theory

Use of conformal mappings in the analysis

Abstract

Suppose $E$ is a subset of the unit circle $\mathbb{T}$ and $H^\infty\subset L^\infty$ is the Hardy subalgebra. We examine the problem of finding the distance from the characteristic function of $E$ to $z^nH^\infty$. This admits an alternate description as a dual extremal problem. Precise solutions are given in several important cases. The techniques used involve the theory of Toeplitz and Hankel operators as well as the construction of certain conformal mappings.