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june 2018 On Weierstrass' monsters in the disc algebra
L. Bernal-González, J. López-Salazar, J.B. Seoane-Sepúlveda
Bull. Belg. Math. Soc. Simon Stevin 25(2): 241-262 (june 2018). DOI: 10.36045/bbms/1530065012

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.

Citation

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L. Bernal-González. J. López-Salazar. J.B. Seoane-Sepúlveda. "On Weierstrass' monsters in the disc algebra." Bull. Belg. Math. Soc. Simon Stevin 25 (2) 241 - 262, june 2018. https://doi.org/10.36045/bbms/1530065012

Information

Published: june 2018
First available in Project Euclid: 27 June 2018

zbMATH: 1407.30031
MathSciNet: MR3819125
Digital Object Identifier: 10.36045/bbms/1530065012

Subjects:
Primary: 30H50
Secondary: 15A03 , 26A27 , 46E10

Keywords: algebrability , disc algebra , lineability , Nowhere differentiable function , spaceability

Rights: Copyright © 2018 The Belgian Mathematical Society

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Vol.25 • No. 2 • june 2018
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