Nowhere Hölderian functions and Pringsheim singular functions in the disc algebra
type: Item
instance of: scholarly article
title: Nowhere Hölderian functions and Pringsheim singular functions in the disc algebra
zbMATH Open document ID: 1445.30028
published in: Monatshefte für Mathematik
publication date: 2019-03-29T00:00:00Z
review text: Let \(\mathbb{D}\) denote the open unit disc and and \(\mathbb{T}\) the unit circle in the complex plane. The disc algebra \(A(\mathbb{D})\) is the uniform algebra of functions continuous on the closed unit disc \(\overline{\mathbb{D}}\) such that \(f|_\mathbb{D}\) is holomorphic. Moreover, \(A^\infty(\mathbb{D})\) is the F-space of all \(f\) in the disc algebra that are smooth on the boundary, that is, all derivatives \(f^{(n)}\) extend continuously to \(\overline{\mathbb{D}}\), endowed with the topology of uniform convergence of all derivatives on \(\overline{\mathbb{D}}\). The authors consider the set \(\mathcal{NH}(\mathbb{T})\) of functions \(f \in A(\mathbb{D})\) which are nowhere hölderian on \(\mathbb{T}\) as well as the set \(\mathcal{PS}(\mathbb{T})\) of all \(f \in A^\infty(\mathbb{D})\) such that \(f|_\mathbb{T}\) is Pringsheim-singular on \(\mathbb{T}\). Here, Pringsheim-singular at \(\zeta =e^{is}\in \mathbb{T}\) means that the Taylor series of \(t \mapsto f(e^{it})\) with center \(s\) has vanishing radius of convergence. The main results state that \(\mathcal{NH}(\mathbb{T})\) and \(\mathcal{PS}(\mathbb{T})\) are strongly \(\mathfrak{c}\)-algebrable as well as \(\mathfrak{c}\)-dense lineable, that is, dense linear subspaces (of dimension \(\mathfrak{c}\)) as well as infinitely generated subalgebras of the disc algebra and \(A^\infty(\mathbb{D})\) respectively exist with the property that all non-zero functions belong to \(\mathcal{NH}(\mathbb{T})\) and \(\mathcal{PS}(\mathbb{T})\) respectively. Moreover, in both cases the existence of corresponding infinite-dimensional Banach subspaces is shown.
Mathematics Subject Classification ID: 30H50
Mathematics Subject Classification ID: 26A27
Mathematics Subject Classification ID: 26E10
Mathematics Subject Classification ID: 46E10
zbMATH DE Number: 7045081
zbMATH Keywords: nowhere Hölderian function
zbMATH Keywords: Pringsheim singular function
zbMATH Keywords: disc algebra
zbMATH Keywords: lineability
zbMATH Keywords: spaceability
zbMATH Keywords: algebrability
author: L. Bernal-González
author: A. Bonilla
author: Jerónimo López-Salazar
author: J. B. Seoane-Sepúlveda
reviewed by: Jürgen Müller
MaRDI profile type: Publication
full work available at URL: s00605-018-1185-8
OpenAlex ID: W2799490088
cites work: Q4177796
cites work: Linear structure of sets of divergent sequences and series
cites work: Lineability
cites work: On dense-lineability of sets of functions on \(\mathbb R\)
cites work: Lineability and spaceability of sets of functions on $\mathbb {R}$
cites work: Algebrability of the set of non-convergent Fourier series
cites work: Nonseparable spaceability and strong algebrability of sets of continuous singular functions
cites work: Über die Baire'sche Kategorie gewisser Funktionenmengen
cites work: Strong \(\mathfrak c\)-algebrability of strong Sierpiński-Zygmund, smooth nowhere analytic and other sets of functions
cites work: Large function algebras with certain topological properties
cites work: Algebrability and nowhere Gevrey differentiability
cites work: Algebras in sets of queer functions
cites work: Functions with big and small successive derivatives everywhere
cites work: Linear Kierst–Szpilrajn theorems
cites work: Lineability of sets of nowhere analytic functions
cites work: Algebraic genericity of strict-order integrability
cites work: Vector spaces of non-extendable holomorphic functions
cites work: Large algebras of singular functions vanishing on prescribed sets
cites work: Highly tempering infinite matrices
cites work: On Weierstrass' monsters in the disc algebra
cites work: Lineability criteria, with applications
cites work: Linear subsets of nonlinear sets in topological vector spaces
cites work: Self-similar functions, fractals and algebraic genericity
cites work: Basic sequences and spaceability in \(\ell_p\) spaces
cites work: Differentiable, Nowhere Analytic Functions
cites work: On functions that are almost continuous and perfectly everywhere surjective but not Jones. Lineability and additivity
cites work: Lineability, spaceability, and additivity cardinals for Darboux-like functions
cites work: Prehistory of Faà Â di Bruno's Formula
cites work: Most Infinitely Differentiable Functions are Nowhere Analytic
cites work: Topological genericity of nowhere differentiable functions in the disc algebra
cites work: Topological genericity of nowhere differentiable functions in the disc and polydisc algebras
cites work: Generic results in classes of ultradifferentiable functions
cites work: Q4488051
cites work: Lineability and additivity in \(\mathbb R^{\mathbb R}\)
cites work: A Characterization of Continuity Revisited
cites work: An undecidable case of lineability in \(\mathbb R^{\mathbb R}\)
cites work: Infinite dimensional Banach spaces of functions with nonlinear properties
cites work: Q5577487
cites work: Q4031612
cites work: On the boundedness of the differentiation operator between weighted spaces of holomorphic functions
cites work: Isometrical embeddings of separable Banach spaces into the set of nowhere approximatively differentiable and nowhere Hölder functions
cites work: The Prevalence of Continuous Nowhere Differentiable Functions
cites work: On Weierstrass' monsters and lineability
cites work: On Weighted Spaces of Harmonic and Holomorphic Functions
cites work: Sur les fonctions non dérivables
cites work: Unendlich oft differenzierbare nicht‐analytische Funktionen
cites work: Q3874454
cites work: Nowhere analytic \(C^\infty\) functions
cites work: Every Separable Banach Space is Isometric to a Space of Continuous Nowhere Differentiable Functions
cites work: Q4340161
cites work: Singularitäten unendlich oft differenzierbarer Funktionen
cites work: Q3137330
DOI: 10.1007/S00605-018-1185-8