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   Common Universal Meromorphic Functions for Translation and Dilation MappingsMeyrath, Thierry  in Computational Methods and Function Theory (in press) We consider translation and dilation mappings acting on the spaces of meromorphic functions on the complex plane and the punctured complex plane, respectively. In both cases, we show that there is a dense ... [more ▼] We consider translation and dilation mappings acting on the spaces of meromorphic functions on the complex plane and the punctured complex plane, respectively. In both cases, we show that there is a dense $G_{\delta}$-subset of meromorphic functions that are common universal for certain uncountable families of these mappings. While a corresponding result for translations exists for entire functions, our result for dilations has no holomorphic counterpart. We further obtain an analogue of Ansari’s Theorem for the mappings we consider, which is used as a key tool in the proofs of our main results. [less ▲] Detailed reference viewed: 22 (0 UL)On a functional-differential equation of A. F. Beardon and functional-differential equations of Briot-Bouquet type; Tomaschek, Jörg in Computational Methods and Function Theory (2013), 13(3), 383-395 We look for solutions of the functional-differential equation f(φ(z))=a(z)f(z)f′(z) for given series φ and a . We show that all formal solutions f of this equation are local analytic. In order to do this ... [more ▼] We look for solutions of the functional-differential equation f(φ(z))=a(z)f(z)f′(z) for given series φ and a . We show that all formal solutions f of this equation are local analytic. In order to do this we transform the equation to a special type of Briot–Bouquet differential equation. [less ▲] Detailed reference viewed: 104 (2 UL)Universal rational expansions of meromorphic functionsMeyrath, Thierry in Computational Methods and Function Theory (2011), 11(1), 317-324 Motivated by known results about universal Taylor series, we show that every function meromorphic on a domain $G$ can be expanded into a series of rational functions, whose partial sums have universal ... [more ▼] Motivated by known results about universal Taylor series, we show that every function meromorphic on a domain $G$ can be expanded into a series of rational functions, whose partial sums have universal approximation properties on arbitrary compact sets $K \subset G^c$. [less ▲] Detailed reference viewed: 109 (4 UL) |
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