Approximation by overconvergence of a power series

Universal Taylor series

“…-Theorem 2.6 strengthens a result of Chui and Parnes [2]. The difference between condition (i) of Definition 1.1 and the assumption, which was used in the paper [2], is that in (i) the compact set K may meet the unit circle, whereas in [2] the hypothesis is : K C {z € C : \z\ > 1}.…”

“…For the construction of an appropriate counterexample we use a strengthened version of a result of Chui and Parnes concerning approximation by overconvergence (cf. [2] It is also true that every Taylor series with radius of convergence greater than or equal to 1, can be expressed as the sum of two universal Taylor series.…”

Coincidence of some classes of universal functions

“…Ostrowski gaps were successfully used, for example in [2,4,6,8,21,22,24,27], to obtain certain properties of universal Taylor series with respect to overconvergence. Here, in order to prove our results, Ostrowski gaps will also be our main tool.…”

“…He proved that there exists a sequence of complex numbers (a n ) such that, for any compact set K ⊂ C with 0 / ∈ K and connected complement and for any function h, continuous on K and holomorphic in the interior of K, there exists a sequence of natural numbers (λ n ) such that zero. Later on, universal Taylor series with strictly positive radius of convergence were defined, first independently by Luh [20] and Chui and Parnes [8] and finally, in their strongest sense, by Nestoridis [28]. Moreover, Melas and Nestoridis have studied in [24] the existence of universal Taylor series with respect to summability methods (see also [5,7]).…”

Universal Taylor series for non-simply connected domains

“…(The corresponding result, where K is required to be disjoint from , had previously been established by Luh [12] and Chui and Parnes [4].) In fact, Nestoridis showed that possession of such universal Taylor series expansions is a generic property of holomorphic functions on simply connected domains , in the sense that U ( ; ) is a dense G subset of the space of all holomorphic functions on endowed with the topology of local uniform convergence (see also Melas and Nestoridis [14] and the survey of Kahane [11]).…”