Interpolation by hypercyclic functions for differential operators
In 1929, \textit{G. D. Birkhoff} [C. R. Acad. Sci. Paris 189, 473--475 (1929; JFM 55.0192.07)] proved the existence of an entire function \(f\) which is universal in the sense that the sequence \(\{f(\cdot +an ):n\geq 1\}\) of its translates (\(a\in \mathbb{C} \backslash \{0\}\)) is dense in the space \(H(\mathbb{C})\) of all entire functions. Later, in 1952, \textit{G. R. MacLane} [J. Anal. Math. 2, 72--87 (1952; Zbl 0049.05603)] demonstrated the same density property for the sequence \(\{f^{(n)}:n\geq 1\}\) of derivatives of some entire function \(f\). Formally speaking, the translation operator \(\tau _{\alpha }:f\mapsto f(\cdot +a)\) and the derivation operator \(D:f\mapsto f'\) acting on the space \(H(\mathbb{C})\) are hypercyclic. In 1991, \textit{G. Godefroy} and \textit{J. H. Shapiro} [J. Funct. Anal. 98, No. 2, 229--269 (1991; Zbl 0732.47016)] unified both the above results by establishing the hypercyclicity on \(H(\mathbb{C})\) of any nonscalar differential operator \(\varphi (D)\), generated by an entire function \(\varphi \) of exponential type. Birkhoff's theorem and MacLane's theorem correspond to the cases \(\varphi (z)=\exp(az)\), \(\varphi (z)=z\), respectively. In a different direction, the Weierstrass interpolation theorem asserts the existence of holomorphic functions on a domain \(\Omega\) of the complex plane \(\mathbb{C}\) having prescribed values at a sequence of points without accumulation points. Recently, Costakis and Vlachou in a preprint with apparently no further references, and independently \textit{M. Niess} [Analysis, München 27, No. 2--3, 323--332 (2007; Zbl 1132.30303), Complex Var. Elliptic Equ. 53, No. 9, 819--831 (2008; Zbl 1153.30003)] carried out the interesting task of combining both properties of hypercyclicity and interpolation. They proved the existence of a holomorphic function \(f\) that is MacLane-universal and simultaneously takes prescribed values at the points of a given set without accumulation points in \(\Omega\), whenever \(\Omega \) is simply connected. They established as well a number of results concerning interpolation for functions enjoying either Birkhoff universality on \(\mathbb{C}\), or multiplicative universality in \(\mathbb{C}\backslash \{0\}\), or universality in a domain with respect to some similarities \(z\mapsto az+b\), or universality (on multiply connected domains) of their Taylor partial sums. The aim of the paper is to extend the resuls by Costakis-Vlachou and Niess. The authors generalize their assertion to differential operators generated by entire functions \(\varphi \) with subexponential type. An entire function \(\varphi \) is said to be of exponential type if there are real constants \(A,B>0\) such that \(|\varphi (z)|\leq A \exp (B|z|)\), (\(z\in \mathbb{C}\)). \(\varphi \) is said to be of subexponential type whenever, given \(\varepsilon >0\), there is a constant \(A=A(\varepsilon)\in (0, +\infty )\) such that \(|\varphi (z)|\leq A \exp (\varepsilon|z|)\), (\(z\in \mathbb{C}\)).
scientific article; zbMATH DE number 5558975
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