Dense-lineability of sets of Birkhoff-universal functions with rapid decay
type: Item
instance of: scholarly article
title: Dense-lineability of sets of Birkhoff-universal functions with rapid decay
zbMATH Open document ID: 1182.30101
author: W. Luh
author: L. Bernal-González
author: María del Carmen Calderón-Moreno
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-12-10T00:00:00Z
review text: 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 \(f\in M\;\{0\}\).
reviewed by: Sergei V. Rogosin
Mathematics Subject Classification ID: 30K05
Mathematics Subject Classification ID: 30D20
zbMATH DE Number: 5644932
zbMATH Keywords: Birkhoff-universal function
zbMATH Keywords: Arakelian set
zbMATH Keywords: growth of entire functions
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.08.049
OpenAlex ID: W2051046646
cites work: Q5642787
cites work: Entire functions with infinite sets of deficient functions.
cites work: Permissible growth rates for Birkhoff type universal harmonic functions
cites work: Recent developments in harmonic approximation, with applications
cites work: Better than Uniform Approximation on Closed Sets by Harmonic Functions with Singularities
cites work: Lineability and spaceability of sets of functions on $\mathbb {R}$
cites work: Q4023465
cites work: Frequently hypercyclic operators
cites work: Q4839148
cites work: Universality of holomorphic functions bounded on closed sets
cites work: Rate of growth of frequently hypercyclic functions
cites work: Q5833361
cites work: Linear Chaos on Fréchet Spaces
cites work: "Counterexamples" to the harmonic Liouville theorem and harmonic functions with zero nontangential limits
cites work: Frequently hypercyclic operators and vectors
cites work: Q4532387
cites work: Q4007848
cites work: Genericity of wild holomorphic functions and common hypercyclic vectors.
cites work: Topological properties of level-sets of entire functions
cites work: Q3342831
cites work: Mean Periodic Functions: Part I. Varieties Whose Annihilator Ideals Are Principal
cites work: Q3752617
cites work: Birkhoff-functions that are bounded on prescribed sets
cites work: Operators with dense, invariant, cyclic vector manifolds
cites work: Universal families and hypercyclic operators
cites work: Q4652147
cites work: Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution
cites work: Q5526246
cites work: Q4340161
DOI: 10.1016/J.JMAA.2009.08.049
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description: scientific article; zbMATH DE number 5644932