An extremal problem for characteristic functions

Chalendar, Isabelle; Garcia, Stephan; Ross, William; Timotin, Dan

doi:10.1090/tran/6446

An extremal problem for characteristic functions
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by Isabelle Chalendar, Stephan Ramon Garcia, William T. Ross and Dan Timotin PDF

Trans. Amer. Math. Soc. 368 (2016), 4115-4135 Request permission

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.

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