A Rudin-de Leeuw type theorem for functions with spectral gaps (Q1979887)

The goal of this paper is to establish a generalization of the Rudin-de Leeuw theorem on the extreme points of the unit ball of the Hardy space \(H^1\) consisting of analytic functions in the unit disk whose boundary functions have finite \(L^1\)-norm on the boundary \(\mathbb{T}\) of the unit disk; that is, \[\|f\|_1=\frac{1}{2\pi}\int_{\mathbb{T}}|f(z)| |dz|<\infty.\] It is well known that \(H^1\) equals functions \(f\in L^1(\mathbb{T})\) with \(\hat{f}(k)=0\) for negative integers \(k\); here \(\hat{f}(k)\) stands for the Fourier coefficients. Using the notation \[\operatorname{spec} f =\{k\in\mathbb{Z}: \hat{f}(k)\neq 0\},\] we may write \[H^1=\{f\in L^1: \operatorname{spec} f\subset \mathbb{Z}_+\},\] where \(\mathbb{Z}_+\) is the set of non-negative integers.\par Recall that if \(X\) is a normed space, the closed unit ball of \(X\) is denoted by \(\operatorname{ball}(X)\); moreover, an element \(x\in\operatorname{ball}(X)\) is called an \emph{extreme point} if it is not an interior point of any line segment contained in \(\operatorname{ball}(X)\). Note that any such point is a unit-norm vector. \par \textit{K. de Leeuw} and \textit{W. Rudin} [Pac. J. Math. 8, 467--485 (1958; Zbl 0084.27503)] proved that a unit-norm function \(f\in H^1\) is an extreme point of \(\operatorname{ball}(H^1)\) if and only if it is an outer function (see also [\textit{J. B. Garnett}, Bounded analytic functions. Pure and Applied Mathematics, 96. New York etc.: Academic Press, A subsidiary of Harcourt Brace Javanovich, Publishers. (1981; Zbl 0469.30024)] and [\textit{K. Hoffman}, Banach spaces of analytic functions. Reprint of the 1962 original. New York: Dover Publications, Inc. (1988; Zbl 0734.46033)]). The author of the present paper considers a finite set of positive integers \[\mathcal{K}=\{k_1,\ldots,k_M\},\] and defines the subspace \[H^1_{\mathcal{K}}:=\{f\in H^1: \operatorname{spec} f\subset \mathbb{Z}_+\setminus \mathcal{K}\}.\] The purpose of this paper is to characterize the extreme points of \(\operatorname{ball}(H^1_{\mathcal{K}})\) equipped with \(L^1\) norm. The main result of this paper (Theorem 1) states that a unit-norm function \(f\in\operatorname{ball}(H^1_{\mathcal{K}})\) with inner-outer factorization \(f=IF\) is an extreme point if and only if \(I\) is a finite Blaschke product whose degree \(m\) does not exceed \(M\); moreover, a certain block matrix (built by using the outer function \(F\) and the \(m\) zeros of \(I\)) has finite rank.