Analytic Theory of Finite Asymptotic Expansions in the Real Domain. Part II-A: The Factorizational Theory for Chebyshev 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 paper, divided into three parts (Part II-A, Part II-B and Part II-C), contains the detailed 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 . It follows two pre-viously published papers: the first, labelled as Part I, contains the complete (elementary but non-trivial) theory for ; the second is a survey highlighting only the main results without proofs. All the material appearing in §2 of the survey is here reproduced in an expanded form, as it contains all the preliminary formulas necessary to understand and prove the results. The remaining part of the survey—especially the heuristical considerations and consequent conjectures in §3—may serve as a good introduction to the complete theory.

This paper, divided into three parts (Part II-A, Part II-B and Part II-C), contains the detailed 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 . It follows two pre-viously published papers: the first, labelled as Part I, contains the complete (elementary but non-trivial) theory for ; the second is a survey highlighting only the main results without proofs. All the material appearing in §2 of the survey is here reproduced in an expanded form, as it contains all the preliminary formulas necessary to understand and prove the results. The remaining part of the survey—especially the heuristical considerations and consequent conjectures in §3—may serve as a good introduction to the complete theory.

KEYWORDS

Asymptotic Expansions, Formal Differentiation of Asymptotic Expansions, Factorizations of Ordinary Differential Operators, Chebyshev Asymptotic Scales

Asymptotic Expansions, Formal Differentiation of Asymptotic Expansions, Factorizations of Ordinary Differential Operators, Chebyshev Asymptotic Scales

Cite this paper

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. doi: 10.4236/apm.2015.58045.

Granata, A. (2015) Analytic Theory of Finite Asymptotic Expansions in the Real Domain. Part II-A: The Factorizational Theory for Chebyshev Asymptotic Scales.

References

[1] 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

[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. (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

[4] 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

[5] 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

[6] Ostrowski, A.M. (1976) Note on the Bernoulli-L’Hospital Rule. American Mathematical Monthly, 83, 239-242. http://dx.doi.org/10.2307/2318210

[7] 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

[8] Coppel, W.A. (1971) Disconjugacy. Lecture Notes in Mathematics. Vol. 220, Springer-Verlag, Berlin.

[9] 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

[10] Granata, A. (1980) Canonical Factorizations of Disconjugate Differential Operators. SIAM Journal on Mathematical Analysis, 11, 160-172. http://dx.doi.org/10.1137/0511014

[11] Granata, A. (1988) Canonical Factorizations of Disconjugate Differential Operator-Part II. SIAM Journal on Mathematical Analysis, 19, 1162-1173. http://dx.doi.org/10.1137/0519081

[12] Karlin, S. and Studden, W. (1966) Tchebycheff Systems: With Applications in Analysis and Statistics. Interscience, New York.

[13] Mazure, M.L. (2011) Quasi Extended Chebyshev Spaces and Weight Functions. Numerische Mathematik, 118, 79-108. http://dx.doi.org/10.1007/s00211-010-0312-9

[14] Pólya, G. (1922) On the Mean-Value Theorem Corresponding to a Given Linear Homogeneous Differential Equations. Transactions of the American Mathematical Society, 24, 312-324.

http://dx.doi.org/10.2307/1988819

[15] Schoenberg, I.J. (1982) Two Applications of Approximate Differentiation Formulae: An Extremum Problem for Multiply Monotone Functions and the Differentiation of Asymptotic Expansions. Journal of Mathematical Analysis and Applications, 89, 251-261.

http://dx.doi.org/10.1016/0022-247X(82)90101-9

[1] 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

[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. (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

[4] 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

[5] 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

[6] Ostrowski, A.M. (1976) Note on the Bernoulli-L’Hospital Rule. American Mathematical Monthly, 83, 239-242. http://dx.doi.org/10.2307/2318210

[7] 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

[8] Coppel, W.A. (1971) Disconjugacy. Lecture Notes in Mathematics. Vol. 220, Springer-Verlag, Berlin.

[9] 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

[10] Granata, A. (1980) Canonical Factorizations of Disconjugate Differential Operators. SIAM Journal on Mathematical Analysis, 11, 160-172. http://dx.doi.org/10.1137/0511014

[11] Granata, A. (1988) Canonical Factorizations of Disconjugate Differential Operator-Part II. SIAM Journal on Mathematical Analysis, 19, 1162-1173. http://dx.doi.org/10.1137/0519081

[12] Karlin, S. and Studden, W. (1966) Tchebycheff Systems: With Applications in Analysis and Statistics. Interscience, New York.

[13] Mazure, M.L. (2011) Quasi Extended Chebyshev Spaces and Weight Functions. Numerische Mathematik, 118, 79-108. http://dx.doi.org/10.1007/s00211-010-0312-9

[14] Pólya, G. (1922) On the Mean-Value Theorem Corresponding to a Given Linear Homogeneous Differential Equations. Transactions of the American Mathematical Society, 24, 312-324.

http://dx.doi.org/10.2307/1988819

[15] Schoenberg, I.J. (1982) Two Applications of Approximate Differentiation Formulae: An Extremum Problem for Multiply Monotone Functions and the Differentiation of Asymptotic Expansions. Journal of Mathematical Analysis and Applications, 89, 251-261.

http://dx.doi.org/10.1016/0022-247X(82)90101-9