On Polynomial Sequences With Restricted Growth Near Infinity

Abstract

Let (P n ) be a sequence of polynomials which converges with a geometric rate on some arc in the complex plane to an analytic function. It is shown that if the sequence has restricted growth on a closed plane set E which is non-thin at ∞, then the limit function has a maximal domain of existence, and (P n ) converges with a locally geometric rate on this domain. If (s n k ) is a sequence of partial sums of a power series, a similar growth restriction on E forces the power series to have Ostrowski gaps. Moreover, the requirement of non-thinness of E at ∞ is necessary for these conclusions.

Overconvergent series of rational functions and universal Laurent series

“…In [4,Theorem 1], it was proved that if there exists a closed set E, non-thin at ∞, such that lim sup n→∞ |P n (z)| 1/dn ≤ 1 for all z ∈ E, then f has a maximal domain of existence and the convergence of (P n ) ∞ n=1 extends to that domain. In the present paper, we generalize this result to sequences of rational functions with fixed poles and apply our main theorem to prove that two classes of universal Laurent-type series coincide.…”

“…In the sequel, we use a characterization of non-thinness which was given in [4], for the case w = ∞. Here we need this notion for finite w. The proof can be easily derived from Lemma 1 of [4].…”

“…In [4,Remark 2], it was shown that for any compact set E ⊂ C ∞ which is thin at ∞ (and has connected complement) and for any z 0 ∈ C \ E, there exists a sequence of polynomials P n of respective degrees ≤ n such that, for all…”

Existence of Universal Taylor Series for Nonsimply Connected Domains

“…If the domain U were unbounded, then there would exist a closed set E U , non-thin at in…nity, such that lim sup k!1 jS N k j 1=N k 1 on E. By Theorem 1 of Müller and Yavrian [18], this would then imply that U . Since we have assumed that nU 6 = ;, the domain U must be bounded.…”

Boundary behaviour of functions which possess universal Taylor series

“…Since F is non-thin at in…nity we can apply a result of Müller and Yavrian [15] to see that lim sup k!1 u k < 0 on . Further (cf.…”