Universality and Cesàro Summability Emmanouil Katsoprinakis, Vasilis Nestoridis and Christos Papachristo doulos

Abstract

Let Ω be an arbitrary domain in the complex plane, Ω ≠ ℂ, and ζ ∈ Ω. Let R = dist(ζ, ∂Ω) Ω (0, +∞), C(ζ, R) = {z ∈ ℂ: |z − ζ| = R} and J(Ω, ζ) = ∂Ω ∩ C(ζ, R). If f is a holomorphic function in Ω, then its Taylor series with center at Ω, Σ ^∞ _n=0 c _n (z − ζ)ⁿ, universal with respect to J(Ω, ζ), if the sequence of its partial sums approximates uniformly any continuous function h: K → ℂ, where K is any compact subset of J(ζ, ζ) with connected complement. In this paper, first we prove the existence of such universal functions in Ω. Secondly, if z ₀ ∈ K, we give sufficient conditions on z ₀ and K which guarantee that the above Taylor series of the universal function f is not (C, a)-summable for every a > − 1. Finally, we give examples of Taylor series Σ ^∞ _n=0 c _n z ⁿ converging on $\mathbb{D} = \left\{ {z \in \mathbb{C}:\left| z \right| < 1} \right\}$, which are universal for a finite subset K of {z ∈ ℂ: |z| ≥ 1}, such that the series Σ ^∞ _n=0 c _n z ₀ ⁿ is (C, a)-summable at every point $z_0 \in K \cap \mathbb{T},\mathbb{T} = \left\{ {z \in \mathbb{C}:\left| z \right| = 1} \right\}$, and every a > 1 and some extensions of this.