Grouped power series

Abstract

We consider an arbitrary series of the form Vn~n, k~ ~ N, ~ q-k,~ < %n+a q-kn+a, P~'n (z) a polynomial of degree kn.In particular, if Vn~a qk~< %~+i, then series (i) is a grouped power series, which when the parentheses are expanded corresponds to a power series ~:=o CrnZm"It is well known [i, 2] that sometimes a power series may be so grouped that the resulting power series converges compactly uniformly in some domain outside the disk of convergence of the original powe r series.This phenomenon is called superconvergence of a power series.We solve the following problem.We must find necessary and sufficient conditions on a set E such that compact uniform or pointwise convergence of an arbitrary series (i) on E implies that (i) converges compactly uniformly in C ~, and in particular, such that series (i) is an entire function.It is easy to construct a grouped power series converging compactly uniformly in Cx\ {z ~-x, Vx • 0}, converging compactly uniformly on {z----x, Vx > 0}, and not converging uniformly on any disk {] z I~ R}; in constructing such a series it is essential to use the equall~m~7~Ikn = oo. Therefore, in what follows we assume that series (i) satisfies the ity

On Polynomial Sequences With Restricted Growth Near Infinity

“…Our main result shows that the convergence is spread out in the whole complex plane if (d n+1 /d n ) is bounded, or to the whole domain of existence of f in the general case. Similar questions were considered earlier in [1] and [7] for the case in which the P n are partial sums of power series. The corresponding results are compared with ours in Section 4.…”

“…To see this, we note that from the conditions of [1,Theorem 8] it follows that cap(E R k ) > R 1−ε k k for some sequence R k → ∞ and some sequence 0 < ε k → 0 + , which implies that E is non-thin at ∞ (see Remark 1).…”

“…We remark that step 1 and (18) in the proof of Lemma 2 may also be derived from results of Korovkin (see [6]; see also [1]). …”

“…As we have indicated in Remark 1, there are sets E with cap(E R ) → ∞, which are thin at ∞. For an example we refer to [1], where a power series without any O-gaps is constructed (that is, there are no sequences (p k ) and (q k ) such that lim ν∈∪ k [p k ,q k ] |a ν | 1/ν = 0 and q k /p k → ∞) for which a subsequence of partial sums converges on some set E with cap(E R ) → ∞. According to Theorem 2, such a set is necessarily thin at ∞.…”