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 APM  Vol.5 No.8 , June 2015
Analytic Theory of Finite Asymptotic Expansions in the Real Domain. Part II-C: Constructive Algorithms for Canonical Factorizations and a Special Class of Asymptotic Scales
Abstract: This part II-C of our work completes the factorizational theory of asymptotic expansions in the real domain. Here we present two algorithms for constructing canonical factorizations of a disconjugate operator starting from a basis of its kernel which forms a Chebyshev asymptotic scale at an endpoint. These algorithms arise quite naturally in our asymptotic context and prove very simple in special cases and/or for scales with a small numbers of terms. All the results in the three Parts of this work are well illustrated by a class of asymptotic scales featuring interesting properties. Examples and counterexamples complete the exposition.
Cite this paper: Granata, A. (2015) Analytic Theory of Finite Asymptotic Expansions in the Real Domain. Part II-C: Constructive Algorithms for Canonical Factorizations and a Special Class of Asymptotic Scales. Advances in Pure Mathematics, 5, 503-526. doi: 10.4236/apm.2015.58047.
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. (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.

[3]   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

[4]   Trench, W.F. (1974) Canonical Forms and Principal Systems for General Disconjugate Equations. Transactions of the American Mathematical Society, 189, 319-327.
http://dx.doi.org/10.1090/S0002-9947-1974-0330632-X

[5]   Levin, A.Yu. (1969) Non-Oscillation of Solutions of the Equation . Uspekhi Matematicheskikh Nauk, 24, 43-96; Russian Mathematical Surveys, 24, 43-99.
http://dx.doi.org/10.1070/RM1969v024n02ABEH001342

[6]   Karlin, S. (1968) Total Positivity, Vol. I. Stanford University Press, Stanford, California.

[7]   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

[8]   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

 
 
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