## Bulletin of the Belgian Mathematical Society - Simon Stevin

### On Weierstrass' monsters in the disc algebra

#### 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