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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Error bounds, duality, and the Stokes phenomenon. I


Author: V. P. Gurariĭ
Original publication: Algebra i Analiz, tom 21 (2009), nomer 6.
Journal: St. Petersburg Math. J. 21 (2010), 903-956
MSC (2010): Primary 30E15
Published electronically: September 23, 2010
MathSciNet review: 2604544
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Abstract: We consider classes of functions uniquely determined by coefficients of their divergent expansions. Approximating a function in such a class by partial sums of its expansion, we study how the accuracy changes when we move within a given region of the complex plane. Analysis of these changes allows us to propose a theory of divergent expansions, which includes a duality theorem and the Stokes phenomenon as essential parts. In its turn, this enables us to formulate necessary and sufficient conditions for a particular divergent expansion to encounter the Stokes phenomenon. We derive explicit expressions for the exponentially small terms that appear upon crossing Stokes lines and lead to an improvement in the accuracy of the expansion.


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Additional Information

V. P. Gurariĭ
Affiliation: Mathematics, Faculty of Engineering and Industrial Sciences, Swinburne University of Technology, PO Box 218, Hawthorn 3122, and School of Mathematical Sciences, Monash University, Clayton 3800, VIC, Australia
Email: vgurarii@swin.edu.au; vladimir.gurarii@sci.monash.edu.au

DOI: http://dx.doi.org/10.1090/S1061-0022-2010-01125-7
PII: S 1061-0022(2010)01125-7
Keywords: Asymptotic approximation, error bounds, Stokes phenomenon
Received by editor(s): October 4, 2009
Published electronically: September 23, 2010
Additional Notes: Editorial Note: The following text incorporates changes and corrections submitted by the author after the paper had already been published (in English) in the Russian original of this journal.
Article copyright: © Copyright 2010 American Mathematical Society