Analytic Theory of Finite Asymptotic Expansions in the Real Domain. Part II-B: Solutions of Differential Inequalities and Asymptotic Admissibility of Standard Derivatives
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.
References
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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]
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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
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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
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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
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, 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.
