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Approximation of analytic functions: a method of enhanced convergence

Bruno, Oscar P. and Reitich, Fernando (1994) Approximation of analytic functions: a method of enhanced convergence. Mathematics of Computation, 63 (207). pp. 195-213. ISSN 0025-5718. https://resolver.caltech.edu/CaltechAUTHORS:20181106-080649656

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Abstract

We deal with a method of enhanced convergence for the approximation of analytic functions. This method introduces conformal transformations in the approximation problems, in order to help extract the values of a given analytic function from its Taylor expansion around a point. An instance of this method, based on the Euler transform, has long been known; recently we introduced more general versions of it in connection with certain problems in wave scattering. In §2 we present a general discussion of this approach. As is known in the case of the Euler transform, conformal transformations can enlarge the region of convergence of power series and can enhance substantially the convergence rates inside the circles of convergence. We show that conformal maps can also produce a rather dramatic improvement in the conditioning of Padé approximation. This improvement, which we discuss theoretically for Stieltjes-type functions, is most notorious in cases of very poorly conditioned Padé problems. In many instances, an application of enhanced convergence in conjunction with Padé approximation leads to results which are many orders of magnitude more accurate than those obtained by either classical Padé approximants or the summation of a truncated enhanced series.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1090/S0025-5718-1994-1240654-9DOIArticle
ORCID:
AuthorORCID
Bruno, Oscar P.0000-0001-8369-3014
Additional Information:© 1994 American Mathematical Society. Received by the editor August 5, 1992 and, in revised form, August 30, 1993. We thank the reviewer and Professor W. Gautschi for their valuable comments. The first author gratefully acknowledges support from NSF through grant No. DMS-9200002. This work was partially supported by the Army Research Office and the National Science Foundation through the Center for Nonlinear Analysis.
Funders:
Funding AgencyGrant Number
NSFDMS-9200002
Army Research Office (ARO)UNSPECIFIED
Subject Keywords:Power series, enhanced convergence, Padé approximation, conditioning
Issue or Number:207
Classification Code:1991 Mathematics Subject Classification. Primary 30B10; Secondary 41A21, 41A25
Record Number:CaltechAUTHORS:20181106-080649656
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20181106-080649656
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:90657
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:06 Nov 2018 18:34
Last Modified:03 Oct 2019 20:27

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