An extremal problem for characteristic functions

Abstract

Abstract. Suppose E is a subset of the unit circle T and H ∞ ⊂ L ∞ is the Hardy subalgebra. We examine the problem of finding the distance from the characteristic function of E to z n H ∞ . 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.

The Brown–Halmos theorem for a pair of abstract Hardy spaces

“…In particular, we recall a result by Qiu [38,Lemma 5.1] (see also [8,Theorem 7.2]) saying that, for every measurable set E ⊂ T and an arc γ ⊂ T of the same measure, there exists an inner function u such that u −1 (γ) and E coincide almost everywhere.…”

“…The next result is one of the most important ingredients in our proof. It appeared in[38, Lemma 5.1] and[8, Theorem 7.2]. If E ⊂ T is a measurable set and γ ⊂ T is an arc such that m(E) = m(γ), then there exists an inner function u satisfying u(0) = 0 and such that the sets u −1 (γ) and E are equal almost everywhere.2.5.…”

“…8) belongs to the Hardy space H p (D). Its nontangential boundary values u(e iϑ ) as z → e iϑ exist for a.e.…”