Nowhere hölderian functions and Pringsheim singular functions in the disc algebra

We prove the existence of dense linear subspaces, of infinitely generated subalgebras and of infinite dimensional Banach spaces in the disc algebra all of whose nonzero members are not α -hölderian at any point of the unit circle for any   α > 0 . This completes the recently established result of...

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Bibliographic Details
Published in:Monatshefte für Mathematik Vol. 188; no. 4; pp. 591 - 609
Main Authors: Bernal-González, L., Bonilla, A., López-Salazar, J., Seoane-Sepúlveda, J. B.
Format: Journal Article
Language:English
Published: Vienna Springer Vienna 11.04.2019
Springer Nature B.V
Subjects:
Algebra
Analytic functions
Banach spaces
Mathematical analysis
Mathematics
Mathematics and Statistics
Subspaces
Topology
Lineability
Secondary 15A03
Pringsheim singular function
46E10
Disc algebra
26A27
Nowhere hölderian function
26E10
Spaceability
Primary 30H50
Algebrability
ISSN:0026-9255, 1436-5081
Online Access:Get full text
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Summary:We prove the existence of dense linear subspaces, of infinitely generated subalgebras and of infinite dimensional Banach spaces in the disc algebra all of whose nonzero members are not α -hölderian at any point of the unit circle for any   α > 0 . This completes the recently established result of topological genericity of this kind of functions, as well as the corresponding lineability statements about functions that are nowhere differentiable at the boundary. Topological and algebraic genericity is also studied for the family of boundary-smooth holomorphic functions that are Pringsheim singular at any point of the unit circle.
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ISSN:0026-9255
1436-5081
DOI:10.1007/s00605-018-1185-8