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

Affiliation(s)

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

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

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.

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.

KEYWORDS

Asymptotic Expansions, Canonical Factorizations of Disconjugate Operators, Algorithms for Canonical Factorizations, Chebyshev Asymptotic Scales

Asymptotic Expansions, Canonical Factorizations of Disconjugate Operators, Algorithms for Canonical Factorizations, Chebyshev Asymptotic Scales

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.

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.

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

[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