Holomorphic functions with universal boundary behaviour
The paper contains a number of results concerning the generic boundary behaviour of functions in \(H(\mathbb{D})\), the Fréchet space of functions holomorphic in the unit open disc \(\mathbb{D}\), and of functions in the analytic Wiener algebra \(W^+\). Theorem 1 says that for a dense \(G_\delta\)-set of functions \(f\) in \(H(\mathbb{D})\) the following property holds: Given any compact set \(K\) strictly contained in \(\mathbb{T}:=\partial \mathbb{D}\), any function \(\varphi\) which is continuous on \(K\), any compact set \(L \subset \mathbb{D}\) and any \(\ell \in \mathbb{Z}\), there exists an increasing sequence \((r_n)\) with \(0\le r_n \to 1\) and such that \(f^{(\ell)} (r_n\zeta+(1-r_n)z)\) converges to \(\varphi(\zeta)\) as \(n \to \infty\), uniformly with respect to \((\zeta, z) \in K\times L\). Also, the radial growth of such functions \(f\) and connections to universal Taylor series are investigated. In particular, the functions necessarily have to grow fast near \(\mathbb{T}\), in the sense that \(\sup_\mathbb{D}|f(z)|/v_\phi(z)=\infty\) for all radial weights \(v_\phi(z)=\exp(\phi(1/(1-|z|))\) with \(\int_1^\infty \log\phi(t)t^{-2} dt< \infty\). On the other hand, in Theorem 2 it is shown that the derivatives of functions from the analytic Wiener algebra generically satisfy a weakened kind of universality: There is a residual set of functions in \(W^+\) having the property that for any family \((\varphi_\ell)_{\ell=1}^\infty\) of measurable functions on \(\mathbb{T}\) there exist a sequence \((r_n)\) as above and a set \(E \subset \mathbb{T}\) of full arc length measure such that, for any \(\ell\in \{1,2, \ldots\}\), any \(\zeta \in E\) and any \(z \in \mathbb{D}\), the sequence \(f^{(\ell)}(r_n \zeta +(1-r_n)z)\) tends to \(\varphi_\ell(\zeta)\) as \(n \to \infty\).
scientific article; zbMATH DE number 7187596
scientific article