Nowhere hölderian functions and Pringsheim singular functions in the disc algebra

Abstract

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 $$\alpha $$ -holderian at any point of the unit circle for any $$\alpha >0$$ . 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.