TITLE:
The Factorizational Theory of Finite Asymptotic Expansions in the Real Domain: A Survey of the Main Results
AUTHORS:
Antonio Granata
KEYWORDS:
Asymptotic Expansions, Formal Differentiation of Asymptotic Expansions, Factorizations of Ordinary Differential Operators, Chebyshev Asymptotic Scales
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.5 No.1,
January
12,
2015
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.