Compositionally universal meromorphic functions

Universal radial limits of meromorphic functions in the unit disk

“…Hence, as a consequence of Theorem 1, we obtain that the set of functions in M (D) having this property contains a dense G δ -subset of M (D). In fact, we have the following stronger statement (see [2,3,[11][12][13] for related results): In particular, the set of functions in M (D) having the above properties contains a dense G δ -subset of M (D).…”

Universal sequences of composition operators

“…The topic has been extensively continued in various directions later on, see for e.g. [1,3,4,5,6,13,14,15,17,22].…”

“…Despite the large amount of contributions, it seems that the problem of the existence of holomorphic functions universal for sequences of composition operators have never been studied when the domains G and Ω differ. Let us however mention a very recent paper by Meyrath [17] in which the author gives necessary and sufficient conditions on a sequence (φ n ) n of holomorphic functions from one domain G to an other domain Ω that guarantee the existence of meromorphic functions in H(Ω) such that the set {f • φ n : n ∈ N} is dense in the Fréchet space of meromorphic functions on G. Rather naturally, it appears that no topological conditions are needed on G and Ω. The reason is practically that allowing (universal) approximation by meromorphic functions allows one to "create" holes anywhere in the domain Ω.…”