The Role of Asymptotic Mean in the Geometric Theory of Asymptotic Expansions in the Real Domain

Antonio Granata^*

Department of Mathematics and Computer Science, University of Calabria, Rende (Cosenza), Italy.

DOI: 10.4236/apm.2015.52013                

Abstract

We call “asymptotic mean” (at +∞) of a real-valued function the number, supposed to exist, , and highlight its role in the geometric theory of asymptotic expansions in the real domain of type (*) where the comparison functions , forming an asymptotic scale at +∞, belong to one of the three classes having a definite “type of variation” at +∞, slow, regular or rapid. For regularly varying comparison functions we can characterize the existence of an asymptotic expansion (*) by the nice property that a certain quantity F（t) has an asymptotic mean at +∞. This quantity is defined via a linear differential operator in f and admits of a remarkable geometric interpretation as it measures the ordinate of the point wherein that special curve , which has a contact of order n - 1 with the graph of f at the generic point t, intersects a fixed vertical line, say x = T. Sufficient or necessary conditions hold true for the other two classes. In this article we give results for two types of expansions already studied in our current development of a general theory of asymptotic expansions in the real domain, namely polynomial and two-term expansions.

Keywords

Asymptotic Expansions, Formal Differentiation of Asymptotic Expansions, Regularly-Varying and Rapidly-Varying Functions, Asymptotic Mean

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	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
[2]	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
[3]	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
[4]	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. (For an Enlarged Version with Corrected Misprints see: arxiv.org/abs/1405.6745v1 [mathCA]. http://dx.doi.org/10.1007/s10476-011-0402-7
[5]	Granata, A. (2014) Analytic Theory of Finite Asymptotic Expansions in the Real Domain. Part II: The Factorizational Theory for Chebyshev Asymptotic Scales. Electronically Archived—arXiv: 1406.4321v2 [math.CA].
[6]	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
[7]	Haupt, O. (1922) über Asymptoten ebener Kurven. Journal für die Reine und Angewandte Mathematik, 152, 6-10; ibidem, 239.
[8]	Sanders, J.A. and Verhulst, F. (1985) Averaging Methods in Nonlinear Dynamical Systems. Springer-Verlag, New York.
[9]	Corduneanu, C. (1968) Almost Periodic Functions. Interscience Publishers, New York.
[10]	Faedo, S. (1946) Il Teorema di Fuchs per le Equazioni Differenziali Lineari a Coefficienti non Analitici e Proprietà Asintotiche delle Soluzioni. Annali di Matematica Pura ed Applicata (the 4th Series), 25, 111-133. http://dx.doi.org/10.1007/BF02418080
[11]	Hallam, T.G. (1967) Asymptotic Behavior of the Solutions of a Nonhomogeneous Singular Equation. Journal of Differential Equations, 3, 135-152. http://dx.doi.org/10.1016/0022-0396(67)90011-3
[12]	Hukuhara, M. (1934) Sur les Points Singuliers des équations Différentielles Linéaires; Domaine Réel. Journal of the Faculty of Science, Hokkaido University, Ser. I, 2, 13-88.
[13]	Ostrowski, A.M. (1951) Note on an Infinite Integral. Duke Mathematical Journal, 18, 355-359. http://dx.doi.org/10.1215/S0012-7094-51-01826-1
[14]	Agnew, R.P. (1942) Limits of Integrals. Duke Mathematical Journal, 9, 10-19. http://dx.doi.org/10.1215/S0012-7094-42-00902-5
[15]	Hardy, G.H. (1911) Fourier’s Double Integral and the Theory of Divergent Integrals. Transactions of the Cambridge Philosophical Society, 21, 427-451.
[16]	Hardy, G.H. (1949) Divergent Series. Oxford University Press, Oxford. (Reprinted in 1973)
[17]	Blinov, I.N. (1983) Absence of Exact Mean Values for Certain Bounded Functions. Izvestija Akademii Nauk SSSR. Serija Mathematicheskaja (Moscow), 47, 1162-1181.
[18]	Ditkine, V. and Proudnikov, A. (1979) Calcul Opérationnel. éditions Mir, Moscou.
[19]	Baumgartel, H. and Wollenberg, M. (1983) Mathematical Scattering Theory. Birkhauser Verlag, Berlin.
[20]	Ostrowski, A.M. (1976) On Cauchy-Frullani Integrals. Commentarii Mathematici Helvetici, 51, 57-91. http://dx.doi.org/10.1007/BF02568143
[21]	Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987) Regular Variation. Cambridge University Press, Cambridge. http://dx.doi.org/10.1017/CBO9780511721434
[22]	Hartman, Ph. (1952) On Non-Oscillatory Linear Differential Equations of Second Order. American Journal of Mathematics, 74, 389-400. http://dx.doi.org/10.2307/2372004
[23]	Hartman, Ph. (1982) Ordinary Differential Equations. 2nd Edition, Birkhauser, Boston.
[24]	Giblin, P.J. (1972) What Is an Asymptote? The Mathematical Gazette, 56, 274-284. http://dx.doi.org/10.2307/3617830