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On Polynomial Sequences with Restricted Growth near Infinity
Author(s) -
Müller J.,
Yavrian A.
Publication year - 2002
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609301008803
Subject(s) - mathematics , sequence (biology) , infinity , domain (mathematical analysis) , polynomial , complex plane , series (stratigraphy) , plane (geometry) , power series , function (biology) , limit (mathematics) , mathematical analysis , combinatorics , pure mathematics , geometry , genetics , evolutionary biology , biology , paleontology
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.