The Factorizational Theory of Finite Asymptotic Expansions in the Real Domain: A Survey of the Main Results

ABSTRACT

After studying finite asymptotic expansions
in real powers, we have developed a general theory for expansions of type (*) ,*x *→ *x _{0}* where
the ordered

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) The Factorizational Theory of Finite Asymptotic Expansions in the Real Domain: A Survey of the Main Results.*Advances in Pure Mathematics*, **5**, 1-20. doi: 10.4236/apm.2015.51001.

Granata, A. (2015) The Factorizational Theory of Finite Asymptotic Expansions in the Real Domain: A Survey of the Main Results.

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. (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:1405.6745v1 [math.CA]. http://dx.doi.org/10.1007/s10476-011-0402-7

[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] Ostrowski, A.M. (1976) Note on the Bernoulli-L’Hospital Rule. American Mathematical Monthly, 83, 239-242.

http://dx.doi.org/10.2307/2318210

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

[7] Trench, W.F. (1974) Canonical Forms and Principal Systems for General Disconjugate Equations. Transactions of the American Mathematical Society, 189, 139-327. http://dx.doi.org/10.1090/S0002-9947-1974-0330632-X

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

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

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

[11] 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

[12] 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

[13] 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

[14] 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. (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. (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:1405.6745v1 [math.CA]. http://dx.doi.org/10.1007/s10476-011-0402-7

[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] Ostrowski, A.M. (1976) Note on the Bernoulli-L’Hospital Rule. American Mathematical Monthly, 83, 239-242.

http://dx.doi.org/10.2307/2318210

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

[7] Trench, W.F. (1974) Canonical Forms and Principal Systems for General Disconjugate Equations. Transactions of the American Mathematical Society, 189, 139-327. http://dx.doi.org/10.1090/S0002-9947-1974-0330632-X

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

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

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

[11] 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

[12] 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

[13] 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

[14] 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