Back
 APM  Vol.5 No.1 , January 2015
The Factorizational Theory of Finite Asymptotic Expansions in the Real Domain: A Survey of the Main Results
Antonio Granata*
Show more
Abstract: After studying finite asymptotic expansions in real powers, we have developed a general theory for expansions of type (*) ,x → x0 where the ordered n-tuple forms an asymptotic scale at x0 , i.e. as x → x0, 1 ≤ i ≤ n – 1, and is practically assumed to be an extended complete Chebyshev system on a one-sided neighborhood of x o. As in previous papers by the author concerning polynomial, real-power and two-term theory, the locution “factorizational theory” refers to the special approach based on various types of factorizations of a differential operator associated to . Moreover, the guiding thread of our theory is the property of formal differentiation and we aim at characterizing some n-tuples of asymptotic expansions formed by (*) and n -1 expansions obtained by formal applications of suitable linear differential operators of orders 1,2,…,n-1. Some considerations lead to restrict the attention to two sets of operators naturally associated to “canonical factorizations”. This gives rise to conjectures whose proofs build an analytic theory of finite asymptotic expansions in the real domain which, though not elementary, parallels the familiar results about Taylor’s formula. One of the results states that to each scale of the type under consideration it remains associated an important class of functions (namely that of generalized convex functions) enjoying the property that the expansion(*), if valid, is automatically formally differentiable n-1 times in two special senses.
Keywords: 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.
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

 
 
Top