On Polynomial Sequences with Restricted Growth near Infinity
Identifieur interne : 000B30 ( Istex/Checkpoint ); précédent : 000B29; suivant : 000B31On Polynomial Sequences with Restricted Growth near Infinity
Auteurs : J. Müller [Allemagne] ; A. Yavrian [Arménie]Source :
- Bulletin of the London Mathematical Society [ 0024-6093 ] ; 2002-03.
Abstract
Let (Pn) 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 (Pn) converges with a locally geometric rate on this domain. If (Snk) 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.
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DOI: 10.1112/S0024609301008803
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<front><div type="abstract">Let (Pn) 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 (Pn) converges with a locally geometric rate on this domain. If (Snk) 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.</div>
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