NDLI: Universal Padé approximants and their behaviour on the boundary

Content Provider	Springer Nature Link
Author	Zadik, Ilias
Copyright Year	2016
Abstract	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 $${\varOmega }$$ of the universal function f. In another kind the universal approximation is not required at any point of $$\partial {\varOmega }$$ but in this case the universal function f can be taken smooth on $$\overline{\varOmega }$$ and, moreover, it can be approximated by its Taylor partial sums on every compact subset of $$\overline{\varOmega }$$ . 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 $${\varOmega }$$ is an open set and S, T are two subsets of $$\partial {\varOmega }$$ that satisfy some conditions, then there exists a universal function $$f\in H({\varOmega })$$ which is smooth on $${\varOmega }\cup S$$ and has some Padé approximants that approximate f on each compact subset of $${\varOmega }\cup S$$ and simultaneously obtain universal approximation on each compact subset of $$(\mathbb {C}{\backslash }\overline{\varOmega })\cup T$$ . A sufficient condition for the above to happen is $$\overline{S}\cap \overline{T}=\emptyset $$ , while a necessary and sufficient condition is not known.
Ending Page	193
Page Count	21
Starting Page	173
File Format	PDF
ISSN	00269255
e-ISSN	14365081
Journal	Monatshefte für Mathematik
Issue Number	1
Volume Number	182
Language	English
Publisher	Springer Vienna
Publisher Date	2016-06-14
Publisher Place	Vienna
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Approximation by rational functions Baire theorem Pade approximants Mathematics Chodral metric Universality Approximation in the complex domain
Content Type	Text
Resource Type	Article
Subject	Mathematics

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