Dense-lineability of sets of Birkhoff-universal functions with rapid decay
A subset \(A \subset {\mathbb C}\) is called an Arakelian set if \(A\) is closed in \({\mathbb C}\) and \({\mathbb C}_{\infty}\backslash A\) is connected and locally connected in \({\mathbb C}_{\infty}\). A function \(f\) from the set of entire functions \(E\) is called a Birkhoff-universal function if the family of its translates \(\{f(a + \cdot): a \in {\mathbb C}\}\) is dense in \(E\). A rather general statement about the existence of large linear manifolds of Birkhoff-universal functions with rapid decay on certain sets, even under the action of appropriate operators, is proved. A typical theorem is: Theorem. Assume that \(A\) is an unbounded subset of \({\mathbb C}\) and that there exists \(\mu > 0\) satisfying \(B(A, \mu) = \{z\in {\mathbb C}: d(A, z) \leq \mu\} \subset A_0\) for some Arakelian set \( A_0\) with \(\varrho({\mathbb C}\;A_0 = \sup \{r > 0: \exists B(z_0, r) \subset A_0\} = + \infty\). Suppose that \(\varphi, \psi: [0, +\infty) \rightarrow (0, +\infty)\) are functions such that \(\varphi\) is continuous, \(\psi\) is increasing, and \(\varphi\) is integrable on \((1, +\infty)\). Then there exists a linear submanifold \(M = M(\varphi, \psi, \mu) \subset E\) satisfying the following properties: (i) \(M\) in dense in \(E\); (ii) Each non-zero function in \(M\) is a Birkhoff-universal function; (iii) \(\lim\limits_{z\rightarrow\infty,\;z\in A} \exp\left(\varphi(|z|) |z|^{3/2}\right) f(z) = 0\) for all \(f\in M\) and \(\lim\limits_{z\rightarrow\infty\;z\in A}\exp\left(|z|^{\alpha}\right) (\Phi(D)f)(z) = 0\) for all \(f\in M\), all \(\alpha < 1/2\), and all \(\Phi\in E\) with \(\tau(\Phi) < \mu\); (iv) \(\lim\limits_{r\rightarrow\infty} \frac{M_f(r)}{\psi(r)} = +\infty\) for all \(f\in M\;\{0\}\).
scientific article; zbMATH DE number 5644932
scientific article