Universal Overconvergence and Ostrowski-Gaps

Abstract

We prove that certain universality properties of the partial sums force a power series to have Ostrowski-gaps. This has interesting consequences for some classes of universal functions.

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.…”

“…K ∩ Ω = ∅, then we have the definitions of the corresponding classes and Functional Analysis we refer to [14,15]. Properties of the universal functions belonging to these classes can be found in [1,2,4,[9][10][11][12]16,19,[23][24][25][26][27][28]30,31]. For example, 26]).…”

Universal Taylor series for non-simply connected domains

“…Melas [13] (see also Costakis [5]) has shown that U ( ; ) 6 = ; for any 2 whenever Cn is compact and connected, and has asked if U ( ; ) can be empty when Cn is compact but disconnected. On the other hand, Müller, Vlachou and Yavrian [15] have shown, for non-simply connected domains , that thinness of the set Cn at in…nity is necessary for U ( ; ) to be non-empty, and have conjectured that this condition is also su¢ cient. There is clearly a large gap between the results of [13] and [15].…”

“…On the other hand, Müller, Vlachou and Yavrian [15] have shown, for non-simply connected domains , that thinness of the set Cn at in…nity is necessary for U ( ; ) to be non-empty, and have conjectured that this condition is also su¢ cient. There is clearly a large gap between the results of [13] and [15]. Also there has been no known example of a domain and points 1 ; 2 2 such that U ( ; 1 ) 6 = ; and U ( ; 2 ) = ;.…”

“…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]). The situation when is non-simply connected is much less well understood, despite much recent research: see, for example, [2], [3], [5], [6], [7], [9], [13], [15], [19], [22], [23], [24], [25]. Melas [13] (see also Costakis [5]) has shown that U ( ; ) 6 = ; for any 2 whenever Cn is compact and connected, and has asked if U ( ; ) can be empty when Cn is compact but disconnected.…”