Recent zbMATH articles in MSC 30H10 https://zbmath.org/atom/cc/30H10 2022-06-24T15:10:38.853281Z Werkzeug A Rudin-de Leeuw type theorem for functions with spectral gaps https://zbmath.org/1485.30016 2022-06-24T15:10:38.853281Z "Dyakonov, Konstantin M." https://zbmath.org/authors/?q=ai:dyakonov.konstantin-m 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. Reviewer: Ali Abkar (Qazvin) A Schauder basis for $$L_2$$ consisting of non-negative functions https://zbmath.org/1485.46012 2022-06-24T15:10:38.853281Z "Freeman, Daniel" https://zbmath.org/authors/?q=ai:freeman.daniel-h-jun "Powell, Alexander M." https://zbmath.org/authors/?q=ai:powell.alexander-m "Taylor, Mitchell A." https://zbmath.org/authors/?q=ai:taylor.mitchell-a The paper gives an essential contribution to the problem of the existence of a Schauder basis in $$L_p$$-spaces consisting of nonnegative functions. The main result implies that every separable infinite dimensional $$L_2(\mu)$$-space for every $$\varepsilon > 0$$ admits a Schauder basis consisting of nonnegative functions with basis constant at most $$1+\varepsilon$$. Another result asserts the existence of a basic sequence in the positive cone of $$L_p(\mathbb R)$$ whose closed linear span contains a subspace isomorphic to $$L_p$$, $$1<p<\infty$$. Finally, it is shown that, for every $$p \in [1,+\infty)$$ there exists a Schauder frame $$(x_j,f_j)$$ for $$L_p(\mathbb R)$$ such that $$(x_j)$$ is a sequence of translates of a nonnegative function. Reviewer: Mikhail M. Popov (Slupsk) The order of $$L^1$$-approximation by elements of the disc algebra https://zbmath.org/1485.46026 2022-06-24T15:10:38.853281Z "Totik, Vilmos" https://zbmath.org/authors/?q=ai:totik.vilmos \textit{D. Khavinson} et al. [Constr. Approx. 14, No. 3, 401--410 (1998; Zbl 0916.46040)] showed that, given a continuous function $$f$$ on the unit circle $$\mathbb T$$ with $$\|f\|_\infty=1$$ and a function $$G\in H^1(\mathbb D)$$ with $$\|f-G\|_1<\epsilon$$, then there exists $$G^*\in A(\mathbb D)$$ with $$\|G^*\|_\infty\leq 1$$ and $\|f-G^*\|_1\leq C\epsilon \log 1/\epsilon.$ In the paper under review it is proved that $$O(\epsilon \log 1/\epsilon)$$ is the precise order. In fact, if $$f=f_\epsilon$$ is given by $$f=F/|F|$$ where $$F(z)=\exp(\epsilon^2/(1+\epsilon -z))\in A(\mathbb D)$$, then $$\|f-F\|_1\leq C\epsilon^2$$, but for any $$G^*\in A(\mathbb D)$$ with $$\|G^*\|_\infty\leq 1$$, one has $$\|f-G^*\|_1\geq c\; \epsilon^2\log 1/\epsilon$$. Reviewer: Raymond Mortini (Metz) Fredholm generalized composition operators on weighted Hardy spaces https://zbmath.org/1485.47037 2022-06-24T15:10:38.853281Z "Sharma, Sunil Kumar" https://zbmath.org/authors/?q=ai:sharma.sunil-kumar "Gandhi, Rohit" https://zbmath.org/authors/?q=ai:gandhi.rohit "Komal, B. S." https://zbmath.org/authors/?q=ai:komal.b-s Summary: The main purpose of this paper is to study Fredholm generalized composition operators on weighted Hardy spaces.