Universal Padé approximants and their behaviour on the boundary
Author(s)
Zadik, Ilias
Download605_2016_935_ReferencePDF.pdf (214.3Kb)
PUBLISHER_POLICY
Publisher Policy
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
Terms of use
Metadata
Show full item recordAbstract
There are several kinds of universal Taylor series. In one such kind the universal approximation is required at every boundary point of the domain of definition Ω of the universal function f. In another kind the universal approximation is not required at any point of ∂Ω but in this case the universal function f can be taken smooth on [bar over Ω] and, moreover, it can be approximated by its Taylor partial sums on every compact subset of [bar over Ω]. Similar generic phenomena hold when the partial sums of the Taylor expansion of the universal function are replaced by some Padé approximants of it. In the present paper we show that in the case of Padé approximants, if Ω is an open set and S, T are two subsets of Ω that satisfy some conditions, then there exists a universal function f ∈ H (Ω) which is smooth on Ω ∪ S and has some Padé approximants that approximate f on each compact subset of Ω ∪ S and simultaneously obtain universal approximation on each compact subset of (C\[bar over Ω]) ∪ T. A sufficient condition for the above to happen is [bar over S] ∩ [bar over T] = ∅, while a necessary and sufficient condition is not known.
Date issued
2016-06Department
Sloan School of ManagementJournal
Monatshefte für Mathematik
Publisher
Springer Vienna
Citation
Zadik, Ilias. “Universal Padé Approximants and Their Behaviour on the Boundary.” Monatshefte für Mathematik 182.1 (2017): 173–193.
Version: Author's final manuscript
ISSN
0026-9255
1436-5081