APM  Vol.5 No.8 , June 2015
Analytic Theory of Finite Asymptotic Expansions in the Real Domain. Part II-B: Solutions of Differential Inequalities and Asymptotic Admissibility of Standard Derivatives
Author(s) Antonio Granata*
ABSTRACT
Part II-B of our work continues the factorizational theory of asymptotic expansions of type (*) , , where the asymptotic scale , , is assumed to be an extended complete Chebyshev system on a one-sided neighborhood of x0. The main result states that to each scale of this type it remains as-sociated an important class of functions (namely that of generalized convex functions) enjoying the property that the expansion (*), if valid, is automatically formally differentiable n ? 1 times in the two special senses characterized in Part II-A. A second result shows that formal applications of ordinary derivatives to an asymptotic expansion are rarely admissible and that they may also yield skew results even for scales of powers.

Cite this paper
Granata, A. (2015) Analytic Theory of Finite Asymptotic Expansions in the Real Domain. Part II-B: Solutions of Differential Inequalities and Asymptotic Admissibility of Standard Derivatives. Advances in Pure Mathematics, 5, 481-502. doi: 10.4236/apm.2015.58046.
References
[1]   Granata, A. (2015) Analytic Theory of Finite Asymptotic Expansions in the Real Domain. Part II-A: The Factorizational Theory for Chebyshev Asymptotic Scales. Advances in Pure Mathematics, 5, 454-480.

[2]   Granata, A. (2007) Polynomial Asymptotic Expansions in the Real Domain: The Geometric, the Factorizational, and the Stabilization Approaches. Analysis Mathematica, 33, 161-198.
http://dx.doi.org/10.1007/s10476-007-0301-0

[3]   Granata, A. (2011) Analytic Theory of Finite Asymptotic Expansions in the Real Domain. Part I: Two-Term Expansions of Differentiable Functions. Analysis Mathematica, 37, 245-287.
http://dx.doi.org/10.1007/s10476-011-0402-7

[4]   Granata, A. (2010) The Problem of Differentiating an Asymptotic Expansion in Real Powers. Part I: Unsatisfactory or Partial Results by Classical Approaches. Analysis Mathematica, 36, 85-112.
http://dx.doi.org/10.1007/s10476-010-0201-6

[5]   Granata, A. (2010) The Problem of Differentiating an Asymptotic Expansion in Real Powers. Part II: Factorizational Theory. Analysis Mathematica, 36, 173-218.
http://dx.doi.org/10.1007/s10476-010-0301-3

[6]   Popoviciu T. (1944) Les Fonctions Convexes. Hermann & C éditeurs, Paris.

[7]   Granata, A. (2015) The Factorizational Theory of Finite Asymptotic Expansions in the Real Domain: A Survey of the Main Results. Advances in Pure Mathematics, 5, 1-20.
http://dx.doi.org/10.4236/apm.2015.51001

[8]   Bourbaki, N. (1976) Fonctions d’une Variable Réelle—Théorie élémentaire. Hermann, Paris.

[9]   Walter, M. and Ford, B. (1911) Conditions Suffisantes pour qu’une Fonction Admette un Développement Asymptotique. Bulletin de la Société Mathématique de France, 39, 347-352.

[10]   Aumann, G. and Haupt, O. (1974) Einführung in die reelle Analysis. I: Funktionen einer reellen Veränderlichen. Walter de Gruyter, Berlin. http://dx.doi.org/10.1515/9783110841046

 
 
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