Nowhere Hölderian functions and Pringsheim singular functions in the disc algebra
Let \(\mathbb{D}\) denote the open unit disc and and \(\mathbb{T}\) the unit circle in the complex plane. The disc algebra \(A(\mathbb{D})\) is the uniform algebra of functions continuous on the closed unit disc \(\overline{\mathbb{D}}\) such that \(f|_\mathbb{D}\) is holomorphic. Moreover, \(A^\infty(\mathbb{D})\) is the F-space of all \(f\) in the disc algebra that are smooth on the boundary, that is, all derivatives \(f^{(n)}\) extend continuously to \(\overline{\mathbb{D}}\), endowed with the topology of uniform convergence of all derivatives on \(\overline{\mathbb{D}}\). The authors consider the set \(\mathcal{NH}(\mathbb{T})\) of functions \(f \in A(\mathbb{D})\) which are nowhere hölderian on \(\mathbb{T}\) as well as the set \(\mathcal{PS}(\mathbb{T})\) of all \(f \in A^\infty(\mathbb{D})\) such that \(f|_\mathbb{T}\) is Pringsheim-singular on \(\mathbb{T}\). Here, Pringsheim-singular at \(\zeta =e^{is}\in \mathbb{T}\) means that the Taylor series of \(t \mapsto f(e^{it})\) with center \(s\) has vanishing radius of convergence. The main results state that \(\mathcal{NH}(\mathbb{T})\) and \(\mathcal{PS}(\mathbb{T})\) are strongly \(\mathfrak{c}\)-algebrable as well as \(\mathfrak{c}\)-dense lineable, that is, dense linear subspaces (of dimension \(\mathfrak{c}\)) as well as infinitely generated subalgebras of the disc algebra and \(A^\infty(\mathbb{D})\) respectively exist with the property that all non-zero functions belong to \(\mathcal{NH}(\mathbb{T})\) and \(\mathcal{PS}(\mathbb{T})\) respectively. Moreover, in both cases the existence of corresponding infinite-dimensional Banach subspaces is shown.
scientific article; zbMATH DE number 7045081
scientific article