Bulletin of the Belgian Mathematical Society - Simon Stevin

On Weierstrass' monsters in the disc algebra

L. Bernal-González, J. López-Salazar, and J.B. Seoane-Sepúlveda

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Abstract

Let $\Omega$ be a Jordan domain in the complex plane whose boundary is piecewise analytic, and let $A(\Omega )$ be the algebra of all holomorphic functions on $\Omega$ that are continuous up to the boundary. We prove the existence of dense linear subspaces and of infinitely generated subalgebras in $A(\Omega )$ all of whose nonzero members are, in a strong sense, not differentiable at almost any point of the boundary. We also obtain infinite-dimensional closed subspaces consisting of functions that are not differentiable at any point of a dense subset of the boundary. In the case of the unit disc, those dense linear subspaces can be found with their functions being nowhere differentiable in the unit circle.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 25, Number 2 (2018), 241-262.

Dates
First available in Project Euclid: 27 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1530065012

Mathematical Reviews number (MathSciNet)
MR3819125

Zentralblatt MATH identifier
06916058

Subjects
Primary: 30H50: Algebras of analytic functions
Secondary: 15A03: Vector spaces, linear dependence, rank 26A27: Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives 46E10: Topological linear spaces of continuous, differentiable or analytic functions

Keywords
Nowhere differentiable function disc algebra lineability spaceability algebrability

Citation

Bernal-González, L.; López-Salazar, J.; Seoane-Sepúlveda, J.B. On Weierstrass' monsters in the disc algebra. Bull. Belg. Math. Soc. Simon Stevin 25 (2018), no. 2, 241--262. https://projecteuclid.org/euclid.bbms/1530065012


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